PDF Archive

Easily share your PDF documents with your contacts, on the Web and Social Networks.

Send a file File manager PDF Toolbox Search Help Contact



John Lee Well Testing .pdf



Original filename: John Lee - Well Testing.pdf
Author: 771060 [ SCAN ]

This PDF 1.6 document has been generated by Adobe Acrobat 8.0 Combine Files / Hewlett-Packard Intelligent Scanning Technology, and has been sent on pdf-archive.com on 04/10/2011 at 13:11, from IP address 95.170.x.x. The current document download page has been viewed 32062 times.
File size: 10.8 MB (160 pages).
Privacy: public file




Download original PDF file









Document preview


Contents
Introduction

1

ColllillUilY
Flow

1.

2.

Fluid
1.1
1.2
1.3
1.4
1.5
1.6

Pressure

2.1
2.2
2.3
2.4

4.

5.

6.

Tests

21

Appendix B: Dimensionless
Inlr(>ducli(m

&llId (Ja~

Variables

RaJi&l1 Fill"' (If a Slightly Colllrr~.'i~ihl~
Radi&ll Flow With ColI~lalll BliP

24
26
27
2()

C: Van Everdingen
and Hurst
Solutions
to
Diffusivity
Equations
1lllrlklul'lillll
('(III:.lanl Rale al 1111I~r1i('Ulldal)'.
No Flow Al'ro~~ ()UI~r lillulldal)'
('oll~lalll Rail' al Imll'r 1illllJlllal)' ,

2.M

W~III)&lIII&lg~

.~()

(illl:.I"111

Prl'~~llrl'

2. ()

Pre~~urc l..cvel in Surrounding

('(III.'ilanl

Pr~.'i.'illr~ al 1IIIIcr lillllllll"I)'.

FOml&lIU)n
Rcservoir Umil!i

No l"low

Al'ro~~ ()(lIl'r

2.10

2.11
2.12

Modilic&llions
Modifications

,llIll Slillllllalioli

Tesl

for Ga!ics
for Mullipha~e Flow

Flow Tests
3.1
Introduction
3.2
Pressure Drdwdown Tests
3.3
Multirate Tests
Analysis
of Well Tests
Using Type Curves
4.1
Introduction
4.2
Fundamenlals of Type Curves
4.3
Ramey's Type Curve!i
4.4
McKinley's Type Curve!i
4.5
Gringarten L't ill. Type Curve!i
for Frdclurcd Wcll!i

35
41

44
45

103
I().~
I'lllilll().~
I(~

Appendix

Appendix

al ()Ull'r

106
II)(J
1(ltJ

li'llllillal)'

1117

IilIUlIll,,'Y

11.'\

D: Rock and Fluid
Property Correlations

119

Inlr()(juclion
Psclld(Il'rilical T~mPl..'I.llure alill
Prc~slIrc of l.il/uid IlyJrlll'iln)(llI~
Bubblc-lX)int Prcssur~ of ('l1Id~ Oil
Solution GOR

II Y

63
63
63
64
68

Oil FOmlalU)n Volume Filclor
Compressibility of Und~~illural~d Oil
COInpressibilily of Sillllral~J ('ruu~ ()il
Oil Viscosity
Solubility of Ga~ in Wal~r
Wal~r Fomtalion Volllm~ Fal"lor
Compr~ssibilily (Jr Wal~r ill
Und~r~illurill~J Rl'~~lvlJir~
Cl)mpr~~sihilily or Wall'r in

12()
121
122
124
124
125

a S,IIUrall'd Rl'~l'lv(lir
Wall'r Vi~l'o~ily
P~l'ullocrilicill Prol~11i~~ or GilS
(J.I~-I..IW I)~vialioll F"l"l(lr (/-I"'aclllr)
anll GilS Foml"lillli Vlllulll~ F.Il'llJr
(jil~ Clllllrr~~~ihiliIY
(ja~ Vi~c(l~ily
1:llnllalilln (")lIlrrl'~~ihilily

126
12X
12X

71
76
76

Other
6. I
0.2
6.3
6.4
6.5

89
HI)
X')
91
97
9M

Development
of Differential
Equations
for Flow
in Porous Media
Introduction
Continuity Equation for
Three-Dimcn~u)nal Fk)w

I () I
I().:?
I ().:?

50
50
50
55

Gas Well Testing
5. I
Introduction
5.2
Ba!iic Theory or G&I!i Flow
in Rescrvoir!;
5.3
Flow-Artcr-Flow
Tc!its
5.4
Isochron&ll Tcsls
5.5
Modifi~d I~ol'hrollal Tc~t~
5.0
U!ie of P!i~ulilipres~ure in
Gas Well Tesl Analysis
Well Tests
IlIlr()UllCli'llI
1111~rl~r~lIcl.'
Tl.'~lillg
Pulse Tc!iling
()rill~tcm Tcst~
Wirclinc Fontlalioll Te~l~

I(HI
IIII

Sillgl~-Ph&l~ I:hlw of Slightly
C()\llpr~~~ibl~ Fluid~
Sillgl~-Ph&l~~ G&I~FI(IW
Simull&ln~ou~ Flow of Oil, W"ll'r,

21
21
23

Appendix

r

Buildup

2
2
2
3
13
15
18

lilr 1{"di,,1 I.'IIJ\\'

Introduction
The Ide&ll Buildup Te!il
Aclual Buildup T~~I~
DcvialuJl1~ From A~~umplu)n!i
in Ideal T~~I Th~OI)'
Qualilaliv~ 8~havior of Fi~ld Te~l~
Effecls and Durdlion of Aflertlow
()el~mlillalu)n of Peml~,lbilily

2.5
2.6
2. 7

3.

Flow in Porous Media
Inlr(>duction
Thc Idc&l1Rc~crvoir Modcl
Solutions 10 Diffusivily EI.JU&llion
R&ldius of Invc!iligalion
Principle of Superposition
Homer's Approxim&ltion

":l/II.llillll

I.&lW~

76
77
79
HJ
M5

A:

100
100

Appendix

E:

A General

Appendix

12(J

..
12X
1.11
131
1.~2

Theory

of Well Testing
Appendix

IIY
IIY
IIY

134

F: Use of SI Units in
Well-Testing
Equations

138

G: Answers
Selected

148

to
Exercises

Nomenclature
Bibliography
Author Index
Subject Index

151
154
156
157

I(X)

~Ul~;c",

Introduction
I'

This textbook explains how to use well pressures and
now rates to evaluate the formation surrounding a
tested well. Basic to this discussion is an understanding of the theory of fluid flow in porous
media and of pressure-volume-temperature (PVT)
rt:lation~ for fluiJ ~ystL'm~of practil;:al iI1tere~t. Thi~
book contains a review of these fundamental concepls, largely in summary form.
One major purpose of well testing is to determine
11I~abililY ofa format")n to pr()du~e reservoir Iluius.
Furtller, it is importallL to determine the underlying
reason for a well's productivity. A properly de~igned,
executed, and analyzed well test usually can provide
informal ion about formulion pemleabilily, exlent of
wellbore damage or stimulalion, re~ervoir pres~ure,
and (perhaps) reservoir boundaries and heterogeneities.
The basic test method i~ Lo create a pressure
drawdewn in the well bore; this causes formation
nuids to enter the wellbore. If we measure the flow
rate and the pressure in the well bore during
production or the pressure during a shut-in period
following production, we usually will have sufficient
infor
tion to characlerize the tested well.
?"a
ThIs book beglJ}~ wllh Ii dl~cus~lon of basIc
equations that describe the unsteady-state Ilow of
fluids in porous media. It then moves into

r-

dis~llssions of pressure buildup tests; pressure
drawdown tests; other now tests; type-curve analysis;
gas well tests; interferen~e and pulse te~ts; and
drillstem and wireline formation tests. Fundamental
principles are emphasized in this discussion, and little
I:ffort i~ made 10 bring lhe intended audi~1ceundergraduate pelroleum engineering students -to
Llie frontier~ of tile subje~t. Tliis role is tilled mu~11
better by other publications, such as the Society of
Pelroleum Engineers' monographs on welliestingl,2
und Alberta Energy Re~ourcL's and Con~ervation
Board'~ gas well testing manual.3
Basic equations and examples use engineering
unil~. However, to ~mooth lile expecled transition to
lhe Inti. System of Units (SI) in the petroleum industry, Appendix F Jis~usses lhis unit system and
restates major equations in SI units. In addition,
answers to examples worked out in the text are given
in SI units in Appendix F.
Iteferences
I. Mallhews,C.S,and Russell,
D.G.: PressureB"i/dupandRow
~ests;1' ~Vells.
Monograph
Series.SP~.Da~las
(1967)I.
.
2. Eilrlullgher.R:C. Jr.: AtI"",,('t'S '" II ell Test AnalysIs,
Monograph
Senc£,
SPE,J}illla£(1977)s.
3. 171('uryuntlJ'rut't;('euflhe
'It-!,.,;,,}:
ufGus ~~/('Ils,
IlairdI:dilion,
I Pub.ECRIJ-75-34,
EncrgyRI:£our~c£
andConservalion
Uoard,
Calgary,Atla.(1975).

I;

'

Cllapter 1

Fluid Flow in Porous Media

1.1 Introduction
In this initial chapter on nuid now in porous media,
\\'c hcgin with a discussion of the differential
Cqllation~ t hat are u~~d most often to model un-

oil), we obtain a partial differential
simplifies to
a2p
J ap
cf>JlC ap

~tcady-~tate now.
SImple statements
of these
cqllations are provided in the text; the more tedious

Il'"

a:z+-a=
r
r

r

equation that

ka'
0.<XX>264
t

(1.1)

mathcmatical details are given in Appendix A for the
in~tructor or student who wishes to develop greater
lInderstanding. The equations are followed by a
di~cll,~sionof some of the most useful solutions to
these equations, with emphasis on the exponentialintcgral solution describing radial, unsteady-state
now. An appended discussion (Appendix B) of
dimcnsionless variables may be useful to some
readcrsat this point.
The chapter concludes with a discussion of the
of
sllperposition.
Superposition,
in
radills-of-investigation
concept and ofillustrated
the principle

if we assume that compressibility, c, is small and
independent of pressure; permeability, k, is constant
and isotropic; viscosity, Jl, is independent of
pressure; porosity, cf>,is constant; and that certain
terms in the basic differential equation (involving
pressure gradients squared) are negligible. This
equation is called the diffusivity equation; the term
0.OOO264klcf>Jlc
is called the hydraulic diffusivity and
frequently is given the symbol '7.
Eq. 1.1 is written in terms of field units. Pressure,
feet;
cf>,square
is a fraction;
viscosity,
p,
is inporosity,
pounds per
inch (psi);
distance,Jl,r, isis in
in

mlilt i\vell infinite reservoirs, is used to simulate
simple reservoir boundaries and to simulate variable
rate production histories. An approximate alternative to superposition. Horner's "pseudoprodlldiml time," completes this discussion.
1.2 The Ideal Reservoir Model
To .dcvelop a~alysis and design techniqu~s fo~ \~ell
tCStlllg, we first must make several simplifYing

centipoise; compressibility, c, is in volume per
volume per psi [c=(I/p) (dpldp)]; permeability, k,
is in millidarcies; time, t, is in hours; and hydraulic
diffusivity, '7,has units of square feet per hour.
A similar eqllation can be developed for the.adial
now of a nonideal gas:
I a
a
cf>
a
-a (~ r -£) = 0.000264 k at ( '!),
(1.2)
r r JlZ
Z

a~sumptiOJ1S
about the well and reservoir that we are
nlOdcling. We Ilaturally make no more simplifying
assllillptions thall are absolutely necessary to obtain
simple, useful solutions to equations describing our
sitllal ion -but we obviously can make no fewer

where Z is the gas-law deviation factor.
For simultaneous now of oil, gas, and water,
I a
ap
cf>c
ap
-a(r
a)=
O-()(X)2~ at'
(1.3)
r r
r.
,

assllmptions. These a~sumptions are introduced as
Ilccdcd, to comhine (I) the law of ~onservation of
mass, (2) Darcy's law, and (3) equations of state to

where c, is the total system compressibility,
c =S c +S c ,+S c +c.
(0
0
WM
g P, f

achieve our objectives. This work is only outlined in
'his cllapter; detail is provided in Appendix A and the
Refercnces.
Consider radial now toward a well in a circular
re~crvoir. If we comhine the law of conservation of
ma~~ and Darcy's law for thc isothermal now of
n\lid~ of small alld constant compressibility (a highly
satisfactory model for single-phase now of reservoir

and the total mobility ~, is the sum of the mobilities
of the individual phases:
k
k
k
.~,'= (-.£ + :.:.I.+ ~).
(1.5)
P-o Jlp, P-w
In Eq. 1.4, So refers to oil-phase saturation, Co to
oil-phase compressibility, ,,>,
M'and c M'to water phasc,
S" and c" to p,asphase; and c f is the formation

h~

dcc,

lill..

(1.4)

..lI.~
"

~
,

t

~;.;.

~

FLUID FLOW IN POROUS MEDIA

compressibility.
meability
and

In

to

oil

1J.0 is the

phase;

oil

and

Eq.

1.5,

in lhe

3

ku

i~ the

presence

viscosilY;

k wand

of

k

p.w

and

refer

effe\:live

the
p.

to

per-

other
refer

tte

al1u

phases,
to the

water

where

Jl

gas

phase.

chapter

becau~e

single-phase

porosity

not

to be in Eqs.

1.1 and

1,3

Solutions
section

fll~ivity

to

a slightly

with
(Section

have

1.2and

Equation

use

solutions

to

1.2) uc~~ribing
liquid

some

in

comments

the

Ihe

a porous

on

dif-

Ilow

of

medium.

solutions

it

Eqs.

called

licularly

four

solutions

useful

bounded

in

well

cylindrical

infinite

to

Eq.

resting:

the

reservoir;

reservoir

with

1.1

are

par-

more

solution

the

a well

that

for

solution

considered

a

for

to

be

an

a line

more

for

Assume

discuss

these

marize

in

an infinite

reservoir.

however,

the

assumptions

that

develop

Eq.

1.1:

medium

of

uniform

rock

and

radial
called

fluid

We will
solutions

and

thickness;

properties;

introduce
to Eq. 1.1.

neces~ary

isotropic

to

porous

pressure

Darcy's
negligible

and

sum-

further

qB;

gradients;

to

that

Cylindrical

Solution
boundary

of Eq.
1.1
conditions

realistic

and

assume
into

the

surface

well

factor

(q refers

conditions,

and

that

and

(3) before

there

solution

sand face

to

flow
Pi.

well.

time

wellbore
outer

e'

boundary;

reservoir

most

useful

form

pressure,

is at
of

Pwf'

rock

the

at the

and

fluid

properties.Thesolutioni~1
qBIJ.

Pwf=Pi

[ 21

-141.2-

~
+2E

for

introduced

+ In'

eO --SOllrcl.'
4

- l

"

a2rr2/_.

[J

-\~12/~

l (a lIe
'

efficiency

and

the dimensionless

O )-J2

of

Eq.

1.7,
on

1.7

satisfactory

are
well

that

3.79x

1

(a

convenience.

..(1.6)

than

ll.'ro

well

»)j1
we

variables

have

the

the

1.67

Since

the

to=0.OOO464kl/~p.CI'W'

more

are the roots

Jl(an'eO)Yl(an)-Jl(an)Yl(an'eO)

of

is

solutions

an accurate.

solution

for
For

times

assllmption

IIII.' uc\:uracy
of IIII.' c'llluliull;
948 <P1J.(.'/,;lk.
Ihe
reservoir's
III\.'

possible:
with

for

well

10 be

of

lilllil~
than

.

the

time

a~~umillg

prl.'SSllrC

an error

f

.

Ei(

to

t h e fl ow

-x)

0.6070

can

Ei function,
For

xsO.02,
can

be

(damage)

near

(1.8)

we can use Table
we use Eq.
considered
most
the

wells

be

by

,

the

,II

i~ 110 lulIgt.'r

h
I
t e so utlon

less than

a line

distrihlili0l1

rt.'!;t.'rvuir

x<0.02.

permeability

pressures

these

Ihe

(if
Eq.

(i.e..

plications
in well testing.
In practice.
we find
thal
=0;

when

<plJ.c,,;/k.

~/'C",~.lk,

Since

1.7 clearly
w from

of

exact

equation

Ei(-x)

Eq.

to

0

x>10.9.

implications

conditions,

1111.ll1lteactll1g.
.
f
I
l fi
A urt ler sImp I Icatlon

10.9.

at

and

question:

solution

tllulillt.'

<XS

(psi)

I (hours).

at radius,

Ei-function

Iu afft.'\:1

evaluate

as
Eq.

(1.7)

and

Analysis

hl.'gill

0.02

to

a logical
and

~11..III.t.' rl.':il.'.rvoir,!;u

To

solution

integral.

Ei(-x)=ln(I.78Ix).

2

and

P-Pi

pressure

properties

<PIJ.CI'I~.lk<I<948

approximated

is at

)

at time

boundary

3.79xlo5

is

rate,

that

approximations

or ~ink)
grealer

W~"

reservoir

(i.e..

~p.CI,2

p,

answer

Eq.

~ize

ap-

begins;

Ihe

calculated

.

\

n

the
solution

to

105

lcss

such

-u

must

approximation

useful

~dl',
U

from

shows3

a5 a

more

k

the

pressures

calculated

solution

at a constant

-'- 948

from

idealized

are

this

Line-Source

area

-I
symbols

1.6 is an exact

ever)

i5

i~

it serves

prodllction

or exponential

we

ht)lllldari\.'s
1.

and

the an

(

we examine

is based

'eO='e/rw

where

function

1.6

~ometime~

(3) the

conditions.

= -~

Before

times

2

e-",I/JJ1(u,,'eO)
n-

where.

3

-2!!-.'eO

kh

the Ei

Eq.

radius

of radius,

the

reservoir

at

volume

begins,
flowing
to

this

we

-x)

com-

Eq.

One

With

Ei
new

mo~t

compare

an infinite

x

qB.

in STB/D

formation
with

across

and

rate

reservoir

The

relates

if
rate.

Ei(

may

before

qBp.

two
A

obtained

flow

B is the

production

pressure.

desired

to

(2) the

is no

is

at constant

in a cylindrical

and

uniform

solution
produces

well bore

in RB/STB);

that
we specify
initial
condition.

discusses
is exacl,

radius;

those

(feet)

It

it

produces

P=Pi+70.6~
the

will

in il~ dl.'v\.'lopnl\.'lll,

solutions.

Pi.

obtain

in

to

we

about

C

a well

drains

CX». Under

instead,

1.1,

has zero

~

requires
and
an

(1) a well

r w' is centered

(I)

Reservoir

practical

that

we

be

form

constant-terminal-

Reservoir

well

distance,
Bounded

Eq.

nor

follows.

pressure.

where

nwdl.'

and

~hould

will

complele

facl

a

pha5e

function5

50lutlon

to

1.1 is

law (sometimes
gravity
forces.

as~umptions

the

approximate)

(2) the

,-

Pw/;

which

Cylindrical

uniform

of

Because

solution

(4) the

pressure-independent
small

of

we

should

were

homogeneous

flow;
applicability
laminar
flow);

Before

we

values

Appendix

wilh

proximate

storage

its

lhi5

produce

It

Everdingen-Hurst

colllpletely.

(but

Bessel

imporlant

in

waler

in

sohllion

van

thaI

1.6
of

(Total

equalion5

equation.

the a5~umptioll~

solution.2

Infinite

solutions,

Eq.

most

exaci

all

formalions

this

forms

the

source
state solution;
with zero
and well
the bore
solution
radius;that the
includes
pseudosteadywell bore
a well

use

The

standard

in

an immobile

at

numerical

i~ an

to

rate

are

to

unu\.'r

BI.'S~I.'I fun\:tion~.

compre5~ibility.)
unfamiliar
with

limiting

that,

even

alarmed

putations.

1.3.

There

be

calculate

useful

compressible

We also

"

Dlffu~lvlty

deals

assumed

necessary

','

equation

1.3 as it was

1.2.

,
This

in Eq.

are

is used

oil contain

have formatioll
The reader

a constanl

YI

CI'

Because
the
formation
is considered
compre5sible
(i.e.,
pore
volume
decreases
as pressure
decrea~es),
is not

and

~ompre~sibililY,

1.8;
zero
have

well bore

1.1

for

and

for

for

ap-

reduced
resulting

4

~'"""""

-,
~-

WELL TESTING

--

from drilli'tg or colttplction opcralion~. Many otllcr
wcll~ arc ~ti,nt,lalcu by acidimtion or Itydraillic
fracturing. Eq. I. 7 fail~ to modcl such wcll~ properly;

q/lll.
~/J.f= 141.2'-k-' -111(rflrlt.)
.f '

its derivation holds the explicit assumption of
uniform permeability throughout the drainage area
ofthewelluptothewellbore.
Hawkins4 pointed out
that if the damaged or stimulated zone is con~idered
eqtlivalent to an altered zone of uniform permeability
(kf) and outer raditls (r s)' the additional pressure
drop across this zone (L\IJ.f)can be modeled by the
steady-state radial now equation (see Fig. 1.1). Thus,

TABLE 1.1. -VALUES

qBp.
-141.2~

In(rslr".)

qBp.
=141.2~(k
'

OF THE EXPONENTIAL INTEGRAL,

k
s

-E/(-

-1)ln(rslrw).

(1.9)

x)

-EI ( -x), 0.000 < 0.209, interval -0.001
x

,
'"

-0.17

'.'
...lc

2.0<x<

1

2

3

4

5

6

7

8

9

0-:00 -.;:;;;
0.01
4.038
0.02 3.355
0.03 2.959
0.04 2.681
0.05 2.468
0.06 2.295
0.072.151
0.08 2.027
0.09 1.919
0.10 1.823

6:332
3.944
3.307
2.927
2.658
2.449
2.279
2.138
2.015
1.909
1.814

5:639
3.858
3.261
2.897
2.634
2.431
2.264
2.125
2.004
1.899
1.805

5:235
3.779
3.218
2.867
2.612
2.413
2.249
2.112
1.993
1.889
1.796

~
3.705
3.176
2.838
2.590
2.395
2.235
2.099
1.982
1.879
1.788

~
3.637
3.137
2.810
2.568
2.377
2.220
2.087
1.971
1.869
1.779

4:545
3.574
3.098
2.783
2.547
2.360
2.206
2.074
1.960
1.860
1.770

4:m3.514
3.062
2.756
2.527
2.344
2.192
2.062
1.950
1.850
1.762

4:259
3.458
3.026
2.731
2.507
2.327
2.178
2.050
1.939
1.841
1.754

~
3.405
2.992
2.706
2.487
2.311
2.164
2.039
1.929~
1.832
1.745

0.11
1.737
0.12 1.660
0.131.589
0.14 1.524
0.15
1.464
0.16
1.409
1.358
0.18 1.310
0.19
1.265
0.20
1.223

1.729
1.652
1.582
1.518
1.459
1.404
1.353
1.305
1.261
1.219

1.721
1.645
1.576
1.512
1.453
1.399
1.348
1.301
1.256
1.215

1.713
1.638
1.569
1.506
1.447
1.393
1.343
1.296
1.252
1.210

1.705
1.631
1.562
1.500
1.442
1.388
1.338
1.291
1.248
1.206

1.697
1.623
1.556
1.494
1.436
1.383
1.333
1.287
1.243
1.202

1.689
1.616
1.549
1.488
1.431
1.378
1.329
1.282
1.239
1.198

1.682
1.609
1.543
1.482
1.425
1.373
1.324
1.278
1.235
1.195

1.674
1.603
1.537
1.476
1.420
1.368
1.319
1.274
1.231
1.191

1.667
1.596
1.530
1.470
1.415
1.363
1.314
1.269
1.227
1.187

0.0
+ ~
4.038
3.335
2.959
2.681
2.468
0.1
1.823
1.737
1.660
1.589
1.524
1.464
0.21.2231.1831.1451.1101.0761.0441.0140.9850.9570.931
0.3
0.906
0.882
0.858
0.836
0.815
0.794
0.4
0.702
0.686
0.670
0.655
0.640
0.625

2.295
1.409

2.151
1.358

2.027
1.309

1.919
1.265

0.774
0.611

0.755
0.598

0.737
0.585

0.719
0.572!

0.5
0.6

0.493
0.404

0.483
0.396

0.473
0.388

0.464
0.381

0.334
0.279
0.235
0.198
0.169
0.144

0.328
0.274
0.231
0.195
0.166
0.142

0.322
0.269
0.227
0.192
0.164
0.140

0.316
0.265
0.223
0.189
0.161
0.138

0.106
0.0915
0.0791
0.0685
0.0595
0.0517
0.0450

0.105
0.0902
0.0780
0.0675
0.0586
0.0510
0.0444

0.103
0.0889
0.0768
0.0666
0.0578
0.0503
0.0438

0.102
0.0876
0.0757
0.0656
0.0570
0.0496
0.0432

-Ei(

"

0

-x), O.OO<x< 2.09, Interval = 0.01

0.560
0.454

0.548
0.445

0.536
0.437

0.525
0.428

0.514
0.420

0.503
0.412

0.7
0.374
0.367
0.360
0.353
0.347
0.340
0.8
0.311
0.305
0.300
0.295
0.289
0.284
0.9
0.260
0.256
0.251
0.247
0.243
0.239
1.0
0.219
0.216
0.212
0.209
0.205
0.202
1.1
0.186
0.183
0.180
0.177
0.174
0.172
1.2
0.158
0.156
0.153
0.151
0.149
0.146
1.30.1350.1330.1310.1290.1270.1250.1240.1220.1200.118
1.4
0.116
0.114
0.113
0.111
0.109
0.108
1.5
0.1000 0.0985 0.0971 0.0957 0.0943 0.0929
1.6
0.0863 0.0851 0.0838 0.0826 0.0814 0.0802
1.7
0.0747 0.0736 0.0725 0.0715
0.0705 0.0695
1.8
0.0647 0.0638 0.0629 0.0620
0.0612 0.0603
1.9
0.0562 0.0554 0.0546 0.0539 0.0531
0.0524
2.0
0.0489 0.0482 0.0476 0.0469 0.0463 0.0456

..

10.9, Interval = 0.1

x
0
2 4.sg-x1Q-=~
3 1.30x10-2
4 3.78x10-3
5 1.15x10-3
6 3.60xI0-4
7 1.15xI0-4
8 3.77x10-5
9 1.24x10-5
10 4.15x10-8

1
4.26 x 10~~
1.15x10-2
3.35xI0-3
1.02x10-3
3_21x10-4
1.03x10-4
3.37x10-5
1.11x10-5
3.73x10-6

2
mx1r
1.01x10-2
2.97x10-3
9.08x10-4
2.86xI0-4
9.22x10-5
3.02x10-5
9.99x10-6
3.34)(10-6

3
3:2W0-=2
8.94x10-3
2.64x10-3
8.09x10-4
2.55x10-4
8.24x10-5
2.70x10-5
895)(10-6
3.00x10-6

4
2.84x-1~2
7.89x10-3
2.34x10-3
7.19x10-4
228x10-4
7.36x10-5
2.42x10-5
B02x10-6
2.68x10-6

5
6
7
2.49 x 10 -~ 2.19 x 10 -2 '1:92X~
6.87x10-3 6.16x10-3
5.45xI0-3
2.07x10-3
1.84x10-3
1.~x10-3
6.41x10-4
5.71x10-4
5.09x10-4
2.03x10-4
1.82x10-4
1.62x10-4
6.58x10-5
5.89x10-5 5.26x10-5
2.16x10-5
1.94x10-5
1.73x10-5
7.1Bx10-8
6.44)(10-6 5.77x10-8
2.41)(10-8 2.16x10-8
1.94,<10-6

.Adapl@d'rom Nlsle, RG.: "How To Use The Expon@nlialinleoral," Pel Eng.(~uO. 1956)8171.173.

8
9
1.69 x 10 -2 1.48 x 10 -2
4.82x10-3
4.27x10-2
1.45x10-3
1.29x10-3
4.53x10-4
4.04x10-4
1.45x10-4
1.29x10-4
4.71x10-5
4.21x10-5
1.55x10-5
1.39x10-5
5.17x10-8
4.64x10-8
1.74x10-8
1.56x10-6

rr

FLUID FLOW IN POROUS MEDIA

t

-5-,.:

fonnation the damage extelld~, tll~ larger the
numerical value of s. There is no uPI1Crlimit for ~'.
Some newly drilled wells will not flow at all before
stimulation; for these wells, ks =0 and s-~.
If a
well is stimulated (ks >k), s will be negative, and the
deeper the stimulation, the greater the numeril.:al
value of s. Rarely does a stimulated well have a skin
factor less than -7 or -8, and such skin factors
arise only for wells with deeply penetrating, highly
conductive hydraulic fractures. We should notc
finally that, if a well is neither damaged nor

I

P
~e
S
f?
W

I
r

~
S

W

stimulated (k=ks)'
s;O. We caution the reader that
Eq. 1.10 is best applied qualitatively;
actual wells

r

..Before

F' 1 1 S h
t' f
' , ,
Ig, .-w~II~~r~,'c 0 pressure distribution near

Eq. 1.9 simply states that the pressure drop in the
altered zone is inversely proportional to k rather
than to k and that a correction to the pressur: drop in
this region (which assumed the same permeability, k,
as in the rest of the reservoir) must be made.
Combining Eqs. 1.7 and 1.9, we find thalthe tolal
pressuredrop at the well bore is
pj-Pwf=

-70.6~

(

qBJJ. .

E, -kt

[ (

q BJJ.,

; -70.6 ~

-2

948 tPlJoC
tr~

948.1. c r2
'I'll

E, -kt

)

Beyond the Wellbore Using
the Ei-Function Solution

t w

)

In --.
r
s
w
It is convenient to define a skin factor, s, in terms of
the properties of the equivalent altered zone:

) (~)
( ~k - Iln.

(1.10)

kh

0.72 cp,
0.1 md'_5
.-1
1.5 x 10 pSI
3,000 psi,
,
3,()()()It,
0.5 ft,
1.475 RB/STB,

--

h ; 150 ft,
tP ; 0.23, and
s ; o.
Calculate the reservoir pressure at a radius of I ft
after 3 hours of production; then, calculate the
pressur~ at radii of 10 and 100 ft after 3 hours of
produl.:tlon.
Solution. The Ei function is not an accurate solution

[

-qBJJ.
-70.6-

Il ;
k;
c, ;
Pj ;
r I! ;
rw ;
Bo ;

k

s
rw
Thus, the drawdown is
pj-Pwf-

of Pressures

Problem. A well and reservoir have the following

) ( r s )J

( --Ik

1.J-Calculation

characteristics: The well is producing only oil; it is
producing at a constant rate of 20 STB/D. Data
describing the well and formation are

[(

$--

from tile well is 110laffected by the I.'xi~ll.'l1l.:cof till.'
altered zone. Said another way, we use Eq. I. II to
calculate pressures at the sandface of a well with an
altered zone, but we use Eq. 1.7 to calculate pressures
beyond the altered zone in the formation surrounding the well. We have presented no simple equations
that can be used to calculate pressures for radiu!i, r,
~llchthatrw<r<.rs,butthiswilloffernodifficultic!i
In well test analysIs.
Example

In -.
ks
rw
For r=rw, the argument of the Ei function is sufficiently small after a short time that we can use the
logarithmic approximation; thus, the drawdowl1 i!i
1,688 tPJJ.ctr~,,
k
pj-Pwf- -qBJJ.
-70.6-:In
kh
t,

-2

should point out that an altered zone near a particular well affects only the pressure near that welli.c., the pressure in the unaltered formation away

) +Aps

) ( r s )]

( --Ik

rarely can be characterized exactly by such a simplified model.
leaving the discussion of skin factor, we

(
In

1,688tPJJ.Ct"~

)

kl

]
-2s.

toflowequationsuntilt>3.79XI05tPIlCtr~,,/k.Here,
3

(1.11)
Eq. 1.10 provides some insight into the physical
significance of the sign of the skin factor. If a well is
damaged (ks <k), s will be positive, and the greater
the contrast between ks and k and the deeper into the

79x

.IlC

105tP

~

t w = [(3.79 x 105)(0.23)(0.72)

k
.(1.5x 10-5)(0.5)2]/(0.1)
; 235 <I; 3 hours
..
Thus, we can use Eq. 1.7 with satisfactory accuracy if

6

,,":;,

-WELL

",j""

the re~ervoir

is still

rl'~l'rvoir

act a~ an infinite

will

infinite

acting

TESTING-

at this time.

reservoir

until

The

1 > 948

1/>1(("";
1 k.

of Eq.
time

1.6, which

for

de~cribes

a well centered

pressure

behavior

in a cylindrical

with

reservoir

of

Here,

radius r (" The limiting
form of interest is that which
is valid
for large times, so that the summation
involving
exponentials
and
Bessel
functions
is

948 cf>1Lc,r~
k
= r (948)(0.23)(0.72)

negligible;
after
-qB1L

.(1.5

x

10 -5)(3,000)2

J/0.3

= 211 ,900

this time (I >948

P,vf-Pi-141.2-

hours.

p=p.

I

+ 70.6--qB1L £1.

kl,

(

hours,

=

kl,

P,

kh

)
+In

(70.6)(20)(1.475)(0.72)

Note

=3,000+

!h~t

ferentlatlng
a

~

(0.1)(150)

( r (' )

3

r II'

4

during
Eq.

10-5)(1)2

]

= 3,()()() + (1()(»Ei(-0.007849)
=3,000+

V

100 In [(1.781)(0.007849»)

1 -(948)(0.23)(0.72)(1.5X
.E,

10-5)(10)2

]

.pressure,
-0.7849)

"('

+

volume

by dif-

of the reservoir,

,#"

(1.13)

Pi'
with
average
volume
of the well.

The

volumetric
volume
balance.

material
the Ei
in tIle

the rate of pressure
to the liquid-filled

-~V
-=
c, V

I 00

]

.£)-(948)(0.23)(0.72)(1.5XIO-5)(IOO)2
t.
(0.1)(3)

Solution.

We

Pwf=P+

0.0744 qBt
..1.- 1.-2
h
4>c, r~

of 7R.49, tile Ei
qBp.
now

discuss

the

next ~olution
to the radial diffusivity
equation
that
we will use extensively
in this introduction
to well test
analysi~.
Actually,
this solution
(the pseudosteady~tate~olution)isnotnew.ltissimplyalimitingform

the

RB/D
of fluid for t
of 5.615 qB (1124) cu

(1.14)

in Eq. 1.12,

-141.2-ln

within

5.615 qB(1124)
2
c, ( 7rr (' I,cf>)

Substituting

= 3,()()() psi.

P,

some ap!es.ervoir

average
pressure
within
the
of the well can be found
from
The pressure
decrease
(Pi -p)

=~~~~j~.
cf>c,hr('

= 3,000 + 100 Ei( -78.49)

Ilcrc wc notc tllat for an argul1lcnt
function
is essentially zero.

pressure,

resulting
from
removal
of qB
!lours [a total volume removed
ft] is

of 100 ft,

P~l"ldosteady-State

pore

form of Eq. 1.12 is useful for
It i~volves
replacing
origi!,al

Pi-P=
3 000

find,

1.12,

during
this time period,
is inversely
proportional

drainage

In t!lis calculation,
we find the value of
fll11rt ion from Tablc
1.1, Note, a~ indicated
tablc, that it is a negative quantity.

-

=..?I,-I.

Another
plications.

(0.1 )(3)

= 3,000 + (100)( -0.318)
.drainage
= 2,968 pSI.

p-,

we

pore volume
V p. This result leads to a form of well
te~ting
sometimes
called
reservoir
limits
testing,
which seeks to determine
reservoir size from the rate
of pressure decline in a wellbore
with time.

p = 3,000 + 100

At a radius

period

ct>c,hr('

Thus,

., -decline

of 10 ft,

+ 100 E,(

(1.12)

~ot -c,Vp.
--0.234qB

= 2,573 psi.

= 3,000

1

then

=3,000+(100)(-4.27)

At a radius

p

)

0.000527kl
cf>1Lc,r~

thi~ time

Since the liquid-filled
V p (cubic feet), is

(0.1)(3)

4

0 0 744 B
= --~-=-Li-.

al
.Eil-(948)(0.23)(0.72)(1.5X

l

.-141.2~

P1vf

-948cf>1LC,r2

kl

we can use

3

y+lnrl'n--,
r l'D

or
Thll~, for times ~css than 211,900
Eq. 1.7. At a radius of 1ft,

cf>1(C,r~/k),

( 21 D

0.0744 qBt
-..L-

1.-2
ct>c,h r~

l ( r (' )

kh

---,
r w

3
4

J

or
P-P

B
~
1=141.2~ln(~)--.
w
kh

rH,

3
4

]

(1.15)

I

FLUID FLOW IN POROUS MEDIA

7

Eqs. 1.12 and 1.15 become more useful in practice if
they include a skin factor to account for the fact that
most wells are either damaged or stimulated. For
example, in Eq. 1.15,

B

r
rw

~

kh

P-Pwj=141.2~111(-!.)-

3
4

]

data.

S

-+(Ap)

I

B
-q
IJ.
re
P-Pwj=141.2-ln(-)--+s,
kh
rw
.I.
and

3.

31
(1.16)

4

Corl:

data

from

thl:

wc:lImdll:ate

an

efll:l:tlVC:

well is either damaged or stimulated? What i~ the
apparent skin fal:tor?
Solution.
.
d '"
To
estimate
pro Ul:tlVIlY Index, we use E
'q.
1.19:
q
J= P-Pwj

2
cPlJ.c,re

kh

""

perml:ability to oil of 50 md. Does this imply thatlh~

,

[0.000527 kt

-qBIJ.
P;-Pwj-141.2-

formation volume factor is 1.5RI3/STB.
1. Estimate the productivity index for the tl:~tl:d
well.
2. Estimate formation permeability from thl:~1:

100
= (2,000-1,500)

=0.2 STB/psi-D.
+In(~)rw

~ +s ]
4

(1.17)

Further, we can define an average permeability, kJ'
such that
-qBIJ.

P-Pwj=

141.2kh
J

~ re
3J
In( -,:-) -4
w
-h

-qBIJ.I;
-141.2~~n

( re )
;:

~)
w

kJ=
-4

3

]

=

[ln(~)-~
rw
4

J/[ ln(~)-~+s
rw

]

(1.18)

10

=16md.

-

4

3.. Core.data frequen.t~yprovide a better esti.n!~te
of formation permeability than do permeabilities
derived from the productivity index, particularly for
a well that is badly damaged. Since cores indicate a
permeability of 50 md, we conclude that this well is
damaged. Eq. 1.18 provides a method for estimating
the skin factor s:
k
r
3
s = (k -1)[ In( -!. ) -4]

defined as

J

q

kJII

(

1=_=,...(1.19)
P-Pwj
141.2BIJ.[ln(~)-~]
rw

I

I 000 )-0.75]
(141.2)(0.2)(1.5)(0.5)[ln( iis

+s ,

This average permeability, kJ, proves to have
considerable value in well test analysis, as we shall see
later. Note that for a damaged well, the average
permeability kJ is lower than the true, bulk formalion permeability k; in fact, these quantities are
equal only when the skin factor s is zero. Since we
sometimes estimate the permeability of a well from
productivity-index (PI) measurements, and since the
productivity index J (STB/D/psi), of an oil well is

i

average permeability, kJ' only, which is not
necessarily a good approximation of formation
permeability,k.FromEq.I.19,
141.2 JBIJ.[ ln (
-~4
r

from which,
kJ=k

2. We do not have sufficient information to
estimate formation permc:ability; we can I:all:tllatc:

50
= 16-1
4

rw

-.

)[(Iniis1000 ) -0.75 J

= 16.

.Ihis method does not necessarily provide a good
estimate of formation permeability, k. Thus, there is
! a need for a more complele means of characterizing a
producing well than exclusive use of PI information.

I

flow Equations for Generalized Reservoir Geometry
Eq. 1.16 is limited to a well centered in a cirl:ular
drainage area. A similar equationS models pseudosteady-state flow in more general reservoir shapes:

Examplel.2-AnalystsofWell
From PI Test

.qBIJ.

P-P,vj=141.2kh

[ 1

2/n

( IO.06A )
C

2
Arw

3
--+s,
4

]

Prublem. A well produces JOO STB/D oil at a
measured flowing bottomhole pressure (BHP) of
.were
1,500 pSI.
A recent pressure survey showed that
2 000
L
d .=
average reservoir
pSI. ogs In Icate
..A pressure IS,
a .net sand thickness of 10 ft. The well drains an area
th d .
d
f I 000 ft th b h I

h
A
d
area sq fI " and
C = Shralnage
ape fac tor 'f or SpeCI
fiICd ralnage-area
sh ape and weIII ocat Ion, d Imenslon Iess.

radius is 0.25 ft. Fluid samples indicate that, at
current reservoir pressure, oil viscosity is 0.5 cp and

Values of C A are given in Table 1.2; further explanation of the source of these CA values is given in

..

WI

L

ralnage ra

.

.IUS, re'

0,

.

;

e

ore 0 e

_!

(1.20)

.

.

.
..

~

I

8

WELL
TESTING

TRANSl:NT
REGION

PWI

Pwl
PSElroST[,J)Y-STAT[
REGION

~TEAOY-STATE

REGION

"l.

log t

Fig. 1.2-Flow

t

regions on semilogarithmic

paper.

Chap. 2.
Productivity index, J, can be expressed for general
drainage-area geometry as
0.00708 kh
J= ~
=.
10.06 A -~ +s l
P-Pllf
Bp.

I! (
21n

C

r 2
..1

)

4

Fig. 1.3-Flow

graph.

...
pseudosteady-state region, the reservoir IS modeled
by Eq. 1.20 in the general case or Eqs: 1.15 a~d I: 12
for the special case of a well cente~ed In a cyll.ndrlc~1
reservoir. Eq. 1.12 shows the linear relationship
between Pwf and I durin~ p~eudostea~y-state. Th~s
linear

II'

regions on Cartesian.coordinate

relationship

also

exists

In

generalized

reservoir

.

(1.21)
Other numerical constants tab~lated in Table .1.2
allow us to calculate (I) the maximum elapsed time
during which a re~ervoir is infinite acting (~o that.the
Ei-function
solution can be used); (2) th~ time
required for the p~eudosteady-sta~ solution to
predict pressure drawdown within IOJoaccuracy; ~nd
(3) time required for the pseudosteady-state solution
to be exact.
..drainage
For a given reservoir geometry, the maximum time
a reservoir is infinite acting can be deter!11~nedusing
the entry in the column "Use Infinite-System
Solution With Less Than IOJoError for IDA < ."
Since IDA =0.000264 kllf/1p.c/A, this means that the
time in hours is calculated from
f/1p.c
/A IDA
1<
.the
0.(xx)264 k
Time required for the pseudosteady-state equation
to he accurate wit hin 1"/0can be found from the entry
in the column headed "Less Than IOJoError for
I f)..t >" and the relationship
q.IC AI
I > --~ _/__-1J~!_.opinions
0.()()()264 k
Finally, time required for the pseudosteady-state
equation to be exact is found from the entry in the
coltlmn "Exact for If).t > ."
AI this point, il is Ilelpful to depict graphically Ihe
Ilow regimes that occur in different lime range~.
rigs. 1.2 and 1.3 show BIfP, !'1I:f: in a w~llllowing al
con~l~nl r.ale, pl<;,tled as a function of time on both
logarIthmIc and linear scales.
In the transient region, the reservoir is infinite
acting
andis'a
is modeled
by Eq. 1.11
which
that 1~II:f
linear fllnction
of , log
I. implies
In the

geometries.
At times between the end of the transient region
and the beginning of the pseudosteady-state region,
this is a transition region, sometimes called the latetransient region, as in Figs. 1.2 and 1.3. No simple
equation is available to predict the relationship
between BHP and time in this region. This region is
small (or, for practical purposes nonexistent) for a
well centered in a circular, square, or hexagonal
area, as Table 1.2 indicates. However, f?r a
well off-center in its drainage area, the late-transient
region can span a significant time region, as Table
1.2 also indicates.
Note that the determination of when the transient
region ends or when the pseudosteady-state region
begins is somewhat subjective. For~ example, tl~e
limits on applicability of Eqs. 1.7 and 1.12 (st~ted ~n
text earlier) are not exactly the same as given In
Table 1.2 -but
the difference is slight. Other
authors I consider the deviation from Eq. I~ to be
sufficient for I> 379 f/1p.c
/r~ I k that a late-transient
region exists even for a well centered in a cylindrical
reservoir between this lower limit and an upper limit
of 1,1.l6 f/1/tc/r;lk .These apparently contradictory
are nothing more than different judgments
about when the slightly approximate solutions, Eqs.
1.7 and 1.12, can be considered to be identical to the
exact solution, Eq. 1.6.
These concepts are illustrated in Example 1.3.

Exa/71ple 1.3 -Flow
Analysis in
Generalized
Reservoir
Geometry
..
Problem. I. For each <;,fth.e following rese.rvolr geomelries, calculate
tIme (b)
In hours
for whIch (a)state
the
reservoir
is infinite the
acting;
the pseudosteady

.

FLUID FLOW IN POROUS MEDIA

9

TABLE1.2-SHAPEFACTORS
FORVARIOUS
SINGLE.WELL
DRAINAGE
AREASfo

t

-

Use Infinite System

( 2.2458)

Less Than
1%'DA
Error
for
>

Solution With Less
Than
% Error
for 1'DA
<

3.4538

0.51n -Exact
CA
1.3224

31.6

3.4532

-1.3220

0.1

0.06

0.10

6

27.6

3.3178

-1.2544

0.2

0.07

0.09

/-:7

27.1

3.2995

-1.2452

0.2

0.07

0.09

'"

21.9

3.0865

-1.1387

0.4

0.12

0.08

0.9

0.60

0.015

In Bounded Reservoirs
-~

CA

(:)

31.62

()

L!~

In CA

for IDA>

0.1

O.~

0.10

.

I/){~
~{§

].

0.098

-2.3227

1.5659

c:J

30.8828

3.4302

-1.3106

0.1

0.05

0.09

ffi

12.9851

2.5638

-0.8774

0.7

0.25

0.03

rn

4.5132

1.5070

-0.3490

0.6

0.30

0.025

m

3.3351

1.2045

-0.1977

0.7

0.25

0.01.

21.8369

3.0836

-1.1373

0.3

0.15

0.025.

10.8374

2.3830

-0.7870

0.4

0.15

0.025

4.5141

1.5072

-0.3491

1.5

0.50

0.06

2.0769

0.7309

0:0391

1.7

0.50

0.02

3.1573

1.1497

0.4

0.15

0.005

,.
~

I
l

E=I=~'
Z

E:I~j,
2

E=:I~:~31
2

m.
2

-0.1703

10

~

TABLE 1.2 -SHAPE

~

WELL TESTING

FACTORS FOR VARIOUS SINGLE.WELL

( 2.2458 )
0.51n
In Bounded Reservoirs
EHB

CA

1

In CA

-Exact
CA

for 'DA >

DRAINAGE AREAS1o

Less Than
1 % Errbr
for tOA>

Use Infinite System
Solution With Less
Than 1 % Error
for tOA <

0.5813

-0.5425

0.6758

2.0

0.60

0.02

0.1109

-2.1991

1.5041

3.0

0.60

0.005

2
EEB31
Z
L.

.~I

5.3790

1.6825

-0.4367

0.8

0.30

0.01

2.6896

0.9894

-0.0902

0.8

0.30

0.01

4

E-

t- 31
~

E=I

=~I

0.2318

-1.4619

1.1355

4.0

2.00

0.03

0.1155

-2.1585

1.4638

4.0

2.00

0.01

1.0

0.40

0.025

0.175

0.06

cannot use

~
Eo

19,
4

C-

.~

I

2.3606

0.6589

-0.0249

~

-

In vertic~lIy _fractured reservoirs:

[-oJ"l x//x"

I L:J=

use (r~/L/)2

in place of A/r~ lor ~r_~I?:~r~~_sy~~!~~

2.6541

0.9761

-0.0635

2.0346

0.7104

0.0493

0.175

0.09

cannot use

r-~l

1.9686

0.6924

0.0583

0.175

0.09

cannot use

r~l

1.6620

0.5080

0.1505

0.175

0.09

cannot use

1.3127

0.2721

0.2685

0.175

0.09

cannot use

0.5232

0.175

0.09

cannot use

I

r-'Ojl

I L.:J

I
1L:J

a

.

10

I

r~l

1 L:::J

1
~
[=:!iJ

I[

0.7667

-0.2374

I

In water-drive
(:)

In reservoirs
0

reservoirs
-19.1

2.95

of unknown production
25.0

-1.07

---

-1.20

---

character
3.22

,

.,-

"r""'r~~C
0
r ('1
',ML;
"""",
,.'j.~

~'
:,

--

q

c

I

q

r.1-

'- -AREA = Awb (ft2)

Z
;.

q

Fig.

~Pw

1,4-Schematic
interface.

d(p", -p,)
dl

=

of wellbore

with

moving

liquid/gas

p g dz
144 gc dl

Fig,

(1.24)

1,5-Sch~matic
of wellbore
liquid or gas.

Sub~titutiI1g,
~d
= -q;

TI
.IU~,

dl
5.615

p

of

""

dl

(1.25)

J}efillc a wellbore storage constant,
144
('

A"",,

gc

~=--- 5.615p

g

If

cPlLC,rw

dl D
(1.31)

we

~:~~~?~,
ct>c,',r w

define

(1.32)

dID

a dimensionless

wellbore

storage

con-

stant, C so, as
CsD~0.894CslcPclhr~,

q = q +~d(P,,'-P,).
B

(1.27)

I:or zero or \Inchanging ~\Irface pressure, p, (a major
and not nece~~arilyvalid a~~umption),
24 C
q.~r=q+--!.-~.

then

lq

dl
..~,

.q

X 0OOO264kdn
'
_2 ~

L_2
<l>lLc,hr M' dID

qif=qs

Then,
~f

slngle"phase

.Thus,

"
Cs:
(1.26)

containing

W

0.0373q;B ~

=-

=(q.~f-q)B.

B IL

0.00708 kh

(24)(1"~41 ~: A , ~'-~~

r

~R

Q.sf

d

.B

(1.28)
dl

qf

=q.

dt-'
n
D
.~

--C

v

-."""'"

(1.33)

I

.q;
dr D
For c0l1~tant-rate production
becomes
qs'
~=I-CD-.

.
(

,

O.{)()708kh (Pj -p".)
1',,=
''--,
qjB,c
O.(xx)264 kl
I" = -:;:( :-;2-.
'
'I" /~, '"

r

.l"j~lio",,-.-

.

)

(q(/) = q;], Eq. 1.34

dPD
s

dl

V

)
(

To ullderstand the soluti.ol~ to now prob~ems that
include wellbore storage, It ~~neces~aryto. Introdu~e
dinlen~ionle~s variables, ~imllar to those dl~cus~cd In
Appendix B. Let .qi ~e the su~face !ate at 1.=0 and
introduce the deflnltlon~ of dImensionless tlmc and
dil1lel1~ionlessprcssure:

I 34

135

Eq. 1.35 is the inner boundary condit,ion for the
problem of constant-rate flow of a slIghtly compressible liquid with wellbore storage. Note t~at, for
small C.~Dor for small dpvldlD' qsflq= I (I.e:, the
effect of well bore storage or sand face rate wIll be
negligible).

(1.29)

As a second example, consider a weJlbore (Fig. 1.5)
that c0l1tains a single-phase fluid (liquid or gas) and
that is produced at some surface rate, q. If we.let
V "" be the volume of wellbore open to formatIon

(I 3()

(b~rrel~) and c It'h be the compressibility of th~, fluid
in the well bore (evaluated at wellbore cof,dlllons).
the mass-balance components are (1) rate of nuid

.

-~"

FLUIDFLOWIN POROUSMEDIA

Ib

ll1u~,
t

,I pl'\.'~~ur\.'trull~i\.'llt to r\.'ul:lI tll\.' b'luII,I,lri\.'~ 'II' a
=r~/4

11/

=948q,'
,

~/k

11

te~tedre~ervoir).I"orexample,il'uwclli~,-"\.'lIt\.'r\.'Jill
acyJilldricaldraillageareaofraJius't.,tlll.'ll,~l.'ttillg

1"1'"

Stated another way, in time t, a pressure disturballce reaches a distance r;, which we shall call radius
of investigation, as given by the equation

( --.!!-_ ) Y2.

(1.47)
948 q,1lC,
The radius of investigation given by Eq. 1.47 also
proves to be the distallcl.' a sigllificallt prl.'s~url.'
di~turbance is propagatl'd by produl..'tion or illjl'l..'tiUII
at a constant rate. For example, for the formal ion
wilh pressure distribulions shown in Fig. 1.7. application of Eq. 1.47 yields Ihe following resulls.
r;=

,
(hours)
0,1
t~:g
100.0

rj
-.!!!L
32
~~
t,<XX>

r;='e'
the time required for stabilizati,)II,
found to be
t s = 948 q,1'£'lr; / k.

t,\, i~
(1.4M)

It is no coincidence that thi~ist~letimcat.wlli,-"II
pseudosteady-state now begins (I.e" the tlml.' at
w~lich Eq. 1.12 b~comes an a~~~lr~l~approxi..n,ltiOiI
01 thc l'~,Ict ~~)IUIIOl~tu tIll.' dlllu~lvlty 1.'411,ItIOll).A
\~ord ul C,I~I~IOII:I'l)r l)tl!l.'r l,lr:,.IIII,lgl.'-arl'.,1
~11,lpl.'~,
~lllle to slabllize call be qUltl' c.lIllcrl'IlI, u~ IlIustr,It\.',1
In Example 1.3..
...,
Useful ~s the radlus-of-lnvc~tl~alloll conccpl I~.~WI.'
must caullon the reader Ihat It IS no panacea. 1'lr~I,
we nole that it is exaclly correct only for u
homogeneous, isolropic,
cylindrical
reservoir reservoir helerogeneilies will decrease the aCCllral..'Y
l)1'
Eq. 1.47. Fllrlher, Eq. 1.47 is exal..'l onl~ r'll'
describing Ihe lime the maximum prCSSllre Jislurbance reaches radills r; following an inslanlallel)lls
bllrst of injection inlo or prodllclion from a well.
Exacl lacalion of the radius or investigation becl)mes
less well defined for continuous injeclion or
production at constant rare following a change in
rate. limitations kepI in mind, though, the radillsof-investigation concept can serve us well.

Comparison of these resulls wilh the preSSllre
distributions plotted shows Ihat r; as calculated from
Eq. 1.47 is near the point at which the drawdown in
reservoir pressure caused by producing the well
becomesnegligible.
We also use Eq. 1.47 to calculate the radius of
investigation achieved at any time after any rate
change in a well. This is significant because the
..
distance a transient has moved into a formation is
Example J ,4 -CalculatIon
of RadIus
approximately the distance from the well at which
of Investigation
formation properties are being investigated at a
Problem. We wish to run a now test on an exparticular t.ime in a. well t,est..
,ploratory
well for sufficiently long to ensure thaI the
The radius of Investlgallon has several uses In
well will drain a cylinder of more lhan I,OOO-ft
pres~ur~ transi~nt test analysi~ and design. A
radius. Preliminary well and nllid data anallsis sufqualitative use IS to help explain the shape of a
gests thaI k = 100 md, q, = 0.2, ", = 2 x 10 -psi
-,
pressure build~p or pressure drawdown ~u~ve. For
and I' = 0.5 cpo What length now test appears adexample, a buildup curve may have a dlfflcult-tovisable? Whal now rare do YOll suggest?
inlerpret shape or slope al earliest times when lhe
,.
..
radius of invesligalion is in the zone of altered
Scllll8lCtn. The minimum. length fl,?w lest WOlII~
permeabilily, ks' nearest the wellbore. Or, more
propagale a pressur~ tranSlell1 ,al?proxlmal~ly 2,()()(~It
commonly, a pressure buildup curve may change
fro~l t~e well (twice I~e mmll~lum. radius 0" Inshape al long times when the radius of invesligation
vestlgatlon for safelY). Time required IS
reachesthe gencral vicinilY of a reservoir bollndary
t = 948 q,J'c r? / k
(~uch as a sealing falllt) or some massive rescrvl)ir
1,
/
helerogeneilY. (In practice, we find
Ihal a
(948)(0.2)(0.5)(2 x 10 -S )(2,000)2'
.
heterogeneity

or

boundary

inlluences

pressure

=

100

response in a well when the calculated radius of
invesligation is of the order of twice the dislance to
the heterogeneity.)
The radius-of-investigation concept provides a
guide for well tesl design. For example, we may wanl
10 sample reservoir properties at least 500 ft from a
rested well. How long a tesl shall be run? Six hours?
Twenty-four hours? We are not forced to guess -or
to run a lest for an arbitrary length of time that could
be either too short or too long. Instead, we can use
the radius-of-investigation concepl to estimate Ihe
time required to test to the desired depth in the
formation.

In principle, any now rate would suffice -lime
required to achieve a particular radius of investigation is independent of now rate. In praclice,
we require a now rate sufficiently large that pressure
change with time can be recorded with sufficient
precision to be useful for analysis. What constitutes
sufficient precision depends on the particular
pressure gauge used in the lest.
I. 1 5 Pnnclp I eo f S uperposl ' ( Ion
'

The radius-of-investigation equation also provides
a means of estimating the length of time required to
achieve "stabilized" flow (i.e., the time required for

At th!s point, th~ mos~ useful s.olution to the now
equation, the El-function solullon, appears to be
applicable only for
describing the pressure

---

= 75.8 hours.

..

i
i

I:

16

WEll TESTING

Image
Well

Well A

Actual
Well
L

L

q

rAC

q
\

rAB

Well C

,
No Flow
Boundary

Well B

Fig. 1.8-Multiple.well

system

in infinite

reservoir.

Fig. 1.9-Well

near no.flow boundary Illustrating

use of

imaging.

distribution in an infinite reservoir, caused by the
production of a single well in the reservoir, and. most
restrictive of all, production of the well at constant
rate beginning at time zero, In this section, we
demonstrate how application of the principle of
superposition can remove some of these restrictions,
and we conclude with examination of an approximation that greatly simplifies modeling a
variable-rate well.
For our purposes, we state the principle of
superposition in the following way: The total
pressure drop at any point in a reservoir is the sum of
the pressure drops at that point caused by flow in
each of the wells in the reservoir, The simplest
illustration of this principle is the case of more than
one well in an infinite reservoir. As an example,
consider three wells, Wells A, B, and C, that start to
produce at the same time from an infinite reservoir
(Fig. 1.8), Application of the principle of superposition shows that
(Pi -P"1) lotal al WcllJ\
I W II A
= (p. I -p) d IICOC/\

Our next application of the principle -4)f superposition' is to simulate pressure behavior in bounded
c 'd h
II ' F 19d
reservoIrs. onsl er t e we In Ig, , a Istance, L ,
from a single no-flow boundary (such as a sealing

.

+ (Pi-P)dllcloWcIiB
+ (Pi-P)dllctoWcIIC'
In

t~rm~

of

£,

.'

functIons

fault). Mathematically,
and

"
logarIthmic

ap.

prl'Xlmatlons,
A
(p.I -P "tola
.11at a c 1\
1,688r/>ILC,'M'A2
qABlL In
= -70.6 ~
kt
.f )

l (

Ir

-70.6

-~

this problem is identical to

the problem of a well a dist~nce 2L from an "image"
well (i,e., a well that has the same production history

..

) -A2S" I

shown to be a no-flow boundary -i,e., along this line
the pressure gradient is zero, which means that there
can be no flow. Thus, this is a simple two-well-in-aninfinite-reservoir problem:

( -948r/>ILC"AU2
)
kt

qBIL

PI.- p WJ
.r=-706- .kh

qcBIL .( -948<1>ILC"AC2)...(1.49)

-706~£i
.kh

---£,

kl,
.wllcrc

.

a~ the actual well), The reason this two-well system
simulates the behavior of a well near a boundary is
I
a IIne equi d 1st
' ant between t he two weII scan be

I I W II

-70.6~E;k'

produces; qB' Well B; and qc, Well C. Note that this
equation includes a skin factor for Well A. but does
not include skin factors for Wells Band C. Because
most wells have a nonzero skin factor and because we
are modeling pressure inside the zone of altered
permeability near Well A, we must include its skin
factor. However, the presence of nonzero skin
factors for Wells Band C affects pressure only inside
their zones of altered permeability and has no influence on pressure at Well A if Well A is not within
the altered zone of either W~1I B or Well C.
Using this method, we can treat any number of
wells flowing at constant rate in an infinite-acting
reservoir. Thus, we can model so-called interference
tests, which basically are designed to determine
reservoir properties from the observed response in
one well (such as Well A) to production from one or
more other wells (such as Well B or Well C) in a
reservoir. A relatively modern method of conducting
interference tests, called pulse testing, is based on
these ideas. 10

(1n-k)
1,688r/>lLc,r~ ""kt

(

-948 <l>p.ct(2L)2
kt

)
).

kt.

qA rcfcr~ to the rate at which

Well

A

,.,

(1.50)

FLUID FLOW IN POROUS MEDIA

r'

~

(1

~

°
L

ql

I-,,-tl
.t:ell

L

.[In(~~~t~6')-2S"].

l2

Startillg

uttimc

t -..trodll\.'\.'

=
-

this

)

q 2

well

l'tllls,

q I

th\.'

Il\.'W

total

produ~illg

at

rutc

rate

is

(q2

q2.

-lIl)

W\.'

ill-

starlillg

is still
the

inside

a zone

contribution

of

of

altered

Wcll

2

pcrmeahilily.

to

drawdowll

of

reservoir pressure is

Well 2
II

2,

at time II' so that the total rate after II is the
required q2. Note that total elapsed time sin~e this
well started producing is (I -I I); note further that

I

[~:=~~1~::::
(

II'

a Wcll

Well

kh

2

~

q

= -70.6 Jtql8
---

} ") I

(Ap)1 = (Pj-PII

~=~==::~~J==~=

~

1/

,

II'1 } 2

(AJ1)2 =(jl '-P

l

= -70.6 jJ.(q2-QI)B
kh

[ I I ,688 cPJtc/r;v -~. }

2

.In

1

k(/-/I)
Well

3

Similarly, the contribution of a third well is

~ 3 - q 2)

(Ap)3=(Pj-Pwj)3=-70.6

Fig. t.tO-Produclion schedule for variable-ratewell.

.f In t i ,688 cPjJ.C
/r~

(

l
..-Tllus,
Here agaIn, note

that

whether

the Image

well lIas a

nonzero skin factor is immaterial. Its influence
outside its zone of altered permeability is indep'ende~t of whether.this ~one exist~.
Extensions of the Imaging tecllnlq'le

also ~an hI.'

used, for example, to model (I) pressure dislrib'ltion

challges

r
'tln

[

.In
is changed

to

Q3' The problem that we wish to solve is this: At
some time I> 12' what is the pressure at the sandface
of the well? To solve this problem, we will 'Ise
superposition as before, but, in this case, each well
that contributes to the total pressure drawdown will
be at the same position in the reservoir -the wells
simply will be "turned on" at different times.
The first contribution to a drawdown in reservoir
pressure is by a well producing at rate ql starting at
I=; O. This well, in gcnl.'ral, will be inside a zonl.' of
altered permeability; thus, its contribution to
drawdown of reservoir pressure is

thc

well

with

two

8

kh

lln(

1,688 cPjJ.c
tr~.
:
) -2'i1

kl

[ 1,688 cPjJ.c
/r~I' 1 -~

1

..

k(/-/t)
jJ.(Q3 -Q2)B
-70.6
kl
'

producing wells. To illustrate this application,
consider the case (Fig. 1.10) in which a well produces
rate

for

jJ.(Q2-ql)8
-70.6--~-

most frequently used to estimate average drainagearea pressure from pressure buildup tests.
Our final and most important application of the
superposition principle will be to model variable-rate

the

-~1.
J

1

(Ap) 1+ (Ap) 2 + (Ap) 3
-Jl.Qt
--70.6-:

for a well between two boundaries intersecting at
90°; (2) the pressure behavior of a well between two
parallel boundaries; and (3) pressure behavior for
wells in various locations completely surrounded by
no-flow boundaries in rectangular-shaped reservoirs.
This
has been
completely;
the
studylast
by case
Matthews
el studie?
al. II ISquite
one of
the methods

atrateQrfromtimeOtotime/l;at/l,therateis
changed
to Q2; and at time 12'

kh

k(/-/2)

Ihc total
druwdown
in rate is

Pj -Pwj=

jJ.(Q3-Q2)B

r 1,688 cPjJ.cr2 1
-k(---=-~
I

-~J' )

(1.51)

12

Proceeding in a similar way, we can model an
a\:tual well with dozens of rate changes in its history;
we also can model the rate history for a wcll with a
continuously changing rate (with a sequence of
constant-rate periods at the average rate during the
period) -but,
in many such cases, this use of
s'lperposition yields a lengthy eq'lation, tedious to
use in hand calculations. Note, however, that such an
eq'lation is valid only if Eq. 1.11 is valid for the total
time elapsed since the well begal\ to flow at its initial
rate -i.e., for time I, 'j must be ~ re.
-

-18

WELL TESTING

Example 1.5 -Use
of Superposition
Problem. A nowing well i~ compleled in a reservoir
lhal hasI the following
properties.
.
p.B = 21:32
500RB/STB,
psla

p~eudoproducing lime:
I (hours) =
/'
24 cumulative
well, Np(STB)
mostproduction
recentrate,from
Qla.,(STB/D)

JL = 0.44 cp,
k -= 25 md,
/1 = 43 ft.
c, = 18x 10-6psi-l,and
4> = 0.16.

Then, to model pressure behavior at any point in a
reservoir, we can use the simple equation
2
.-=
-70.6JLQlas1B Ei
-9484>JLC,r
P, P
kh
kIp
.(1.53)

(

)

What will the pressure drop be in a shut-in well 500 ft
from the flowing well when the nowing well has been
shut in for I day following a flow period of 5 days al
300 STB/D?

Two questions arise logically at this point: (I) What
is the basis for this approximation? (2) Under what
conditions is it applicable?
The ba~is for the approximation is not rigorou~,

Solution. We must superimpose the contributions of
two wells because of the rate change:

but intuitive, and is founded on two criteria: (I) If we
use a single rate in the approximation, the clear

p. -p=

-~~

I

+ (q
Now '

[q

kh

2

-q

I

I

Ei ( -948 4>JLc,r2
kl

) Ei l

choice

)

::::-948 4>JLc,r
k(I-II)

I] .Sllggc~ls

most

recent

rate;

such

a rate,

maintained

such

choose of an
that that
the we
product
the effective
rate and production
the productionti.me

lime way,
re~ults
in lhe correcl
cumulative
production. acIn
lhi~
material
balances
will be maintained

,

'= [(948)(0.16)(0.44)(1.8XI0-S)

k

is the

for any significant
determines
the pressure
distribution
nearest period,
the wellbore
and approximately

out lo the radius of investigation achieved with that
rate. (2) Given the single rate to use, intuition

2

948 4>JLCr

.(500)2J/25 = 12.01.
Then '
p. -p = -(70.6)(0.44)( 1.32)
I
(25)(43)

.[ (300) Ei [
't

-12.01

I
curatey..
"
Bul when IS the approximation

adequate? I f we

maintain a most-recent rate for too brief a time
interval, previous rates will playa more important
role in determining the pressure distribution in a
tested
We can
offerrate
twoishelpful
guidelines.
First, reservoir.
if the most
recent
maintained
sufficiently long for the radius of investigation achieved
at this rate to reach the drainage radius of the tested

J

(6)(24)

"

well, then Horner's approximation is always sufficiently accurate.
This
quitea new
conservative,
howevcr.
Second, we
findrule
that,is for
well that,
undergoes a series of rather rapid rate changes, it is
usually sufficient to establish the last constant rate

for at lea~t twice as long as the previous rate. When
+ (0 -300) Ei [

I.

(I 52)

~~

Il
(I )(24) J

IlIcrc is any doubl about whet her these guidelines are
satisfied, the safe approach is to use superpo.\1tion tol
model the production history of the well.

= 11.44[-Ei(-0.0834)+Ei(-0.5)j
= 11.44 (1.989 -0.560)

Exal11ple 1.6 -Application

= 16.35 p~i.

Horller's

1.6 Horner

, s Approximation
..ProlJlem.

In 1951, Hornerl2 reporled an approximalion that
can be used in many cases to avoid the u~e or
superposition in modeling the produclion history or a
variable-rate well. With this approximation, we can
replace the sequence of Ei functions, renecting rate
changes, with a single Ei function that contains a
single producing time and a single producing rate.
The single rare is the most recent nonzero rate at
which the well was produced; we call this rate qla~t
for now. The single producing time i~ found by
dividing cumulalive production from the well by the
most recent rate; we call this producing lime Ip. or

of

Approximation

completion,
is produced
for a ~hortFollowing
time and then
shut in aforwell
a buildup
test.
Theproduclionhistorywasasfollows.
Production
Time TotalProduction
~~~~
-(STB)
:~
S~
26
46
72
68
I. Calculate the pseudoproducing time, tp'
2. Is Horner's approximation adequate for this
case? If not, how should the production history for
thi~ well be simulated?

~

r

!

FLUID FLOW IN POROUS

MEDIA

19

,I
"
'

Suilltiun.
1.

I

Ilow IOllg wolilu il lakt: for tilt: wt:llto ~Iaoili,t: al lIlt:
new rate?
68 STB

l/

=

I"~I

72 hours

24 h
x -~~

day

= 22.7 STB/D.

TIII.'II,

I
!
I
I
I

I
r

of radius for tllis

situation
same graph
the plot
for
a
r..te of on
350the
STOll>.
Is theasradiu~
of devl.'lopl.'d
invt:~tigatk)11
calcillat\.'u from Eq. 1.47 Otfft:d\.'d by I.'llallgl: ill Ilo~'
ralt:'! I)ot:~ tilt: exlrapolOtliol1 of tilt: ~Iraiglll lilll:

(24)( 166)
=
227
= 176 hours.
( .)
2. In this case,

referred to in Exer~ise 1.5~llangl."! Wllal i~ tilt: t:fft:l:t
of illcrt:a~t:d ral~'!
I.M Writ4: all equation simil..r to ~q. 1.49 fur tIll.'
cast: in which Wells, A, 0, and C bcglll to prod lice ,It
uiffl.'rent times from onc ..nolhl:r. What do you

= 72
-=
A//lI:XI-lo-lasl 26

i

plot of pressure vs. logarithm

I = ~4(cumulilli~~~CI~"~ ~:~~)
II
ql..,..STUll)

~/I asl

I

1.7 Suppose the well descr~bed in EX4:rCfSe 1.2
flowed at a rate of 700 STOll> lor 10 days. Prep..rt:..

2.77> 2.
...1.9

Thus, Horner's
approximation
IS probably
ilc.I~ljllalefor this case. It should not be necessaryto
II~I.'~llperposition, whi~h is required when Hofller'~
ilpproximation is not adequate.
L'" .'. "
I'.xercises
1.1 Compare values of Ei (-x) and In (I. 781x)
for the following values of x: 0.01, 0.02, 0.1, and I.
Wllal do you conclude about the accuracy of the
Ill~arilhmic approximation?
About its range of
ilpplicability?
1.2 A well has nowed for 10 days at a rate of 350
Sill/D. Rock and nuid properties include B= 1.13
RIl/STO; Pi = 3,000 psia; II.= 0.5 cp; k = 25 md
(ulliform to wellbore-i.e.,
s=O); h=50
ft;
-S
P
"
'
I-I.
A.=()
16'
and
r
=().333
ft.
"
'> xl()
'1-'"
'i'
.,
II'
l'al..:uIOttepressures at radii of 0.333, I, 10, 100,
l.tKX),and 3, 160 ft, and plot the results as pressure
\). Ih~ logarithm of radius. What minimum drainage
JilJiu~have you assumed in this calculation?
1.3 I:or the well described in Exercise 1.2, plot
pJ~)~llr~in the well bore vs. logarithm of time atlimcs
III' 0.1, I, and 10 days. What minimum drainage
ril,lill~ have you assumed in this calculation?
1.4 Calculate (a) elapsed time required for t~~ EiI'lIllction solution to be valid for the conditions
c.I~~..:rib~d
in Exercise 1.2; (b) time required for the
hl~arilhmic approximation of the Ei function to
ilpply for calculations at the wellbore; and (c) time
Jl.'ljuircdfor the logarithmic approximation to apply
Illr I.'alculalions at a radius of I,O{}O ft. Is the
Illgarilhmic approximation valid by the time the Ei
fullction ilself is a valid solution to the now equation
illlhl: wellboreJ At a ra~!us of .I.O{}O.ft? ..B
1.5 Estimate the radius of Investlg~tlon ac.hlev~d
illll.'r 10 days now time for .the re.servolr ~escflbed In
I:x~r..:ise1.2. Compare. this estlm~te w~th the ~xIJilpolalion to 3,000 pSI of the straight hne passIng
tilruligh radii of 0.333 and 100 ft on the plot of
prc)~urevs. logarithm of radius.
On this plot. how far into the formation has a
)jgllifil.'ant pressure disturbance been propagated?
What is the size of the pressure disturbance at the
radiu~ of investigation calculated from Eq. 1.47?
1.6 If the drainage radius of the well described in
I:xcrl.'ise 1.2 were 3,160 ft, and if the now rate at the
"~II ~llddenly was changed from 350 to 500 STB/D,

..ssumc
when youabout
write the
tllis location
equation?of rest:rvoir boulldarit:~
(a) Suppose a well is250 ft uuewcst of an or 111south trending fault. I'rom pressurc transit:nt tcst~,
the skin factor, s, of this wcll has bcen founu 10 Ot:
5.0. Suppose further that the wt:11ha~ been tlo,,'illg
for 8 days at 350 BID; reservoir and well properlit:s
are those given in Exercise 1.2. Calculate pres~urt:..t
the nowing well.
(b) Suppose there is a shut-in well 500 ft due north
of the producing well. Calculate the pressure at the
shut-in well at the end of8 days.
1.10 A reservoir has the following propcrtics.
.
Pi = 2,500 psla,
B = 1.32 RB/STB,
II. = 0.44 cp,
k = 25md,
h = 43ft,
-6
'-1
d
c{ = 18 x 10
pSI, an
cP = O.16.
In this reservoir, a well is opened 10 now at 250
STUll) for I day. The s~conu Jay its now is illcreased to 450 OlD and thetlliru to 5CX)
OlD. What is
the pressure in a shut-in well 66() ft away after the
third day?
I. I I In Example 1.6, Application of Hoflll:r's
Approximation, what innuence did the 12-hour ~IUtin time have on the calculation? How would the
innuence of this shut-in period have changed had tile
sllut-in period been 120 hours? How do you suggest
tllat the calculation procedllre be modified to take
inlo account long shut-in periods prior to producing
at the final rate?
1.12 Consider a well and formation with the
following properties.
= 1.0 RB/STB
= I 0c
'
~ = 10 ftP,
k = 2S md,
= 0 2
cP -3' ~
.
Pi -,
P~t"-1
C{ :: ~O x ~O PSI.
s:=
';~
r w -I.
t.
The well produced 100 STB/D for 3.0 days, was
shut-in for the next 1.0 day, produced 150 STB/D for
the next 2.0 days, produced 50 STB/D for the next
1.0 day, and produced 200 STB/D for next 2.0 days.

'

I
r

t

w

f ~(a) Calculate
the pseudoproducing
time,
t p.
Compare
this
with
the
actual
total
producing
time.
b C I
I
()

a cu ate

and

plot

the

pressure

dIstributIon

In

the reservoir at the end of 9 days using Horner's
al"proximation.
(c) On the same graph
plot
the pressure
...'
dIstributIon

at the end

of 9 days

...3.
usIng superposItIon.

(d) What do you conclude about the adequacy of
Horner's approximation
in this particular case?
1.13 A well and reservoir have the following
I"roperties
.Arca,"
A = 17.42 X 10(' sq ft (40 acres),
Ii> = 0 2
I'

p. =

,
Cp,

Ct = IOxI0-6psi-l,
k'=
100 md,
" = 10 ft
30 '
S = .,
r,., = 0.3ft,and
B = 12 Rn/STIJ.
For each of the drainage areas in Table 1.2, determine (a) the time (hours) up to which the reservoir is
infinite-acting;
(b) the time (hours) beyond which the
,
.for
pseu?ost~ady-state
solutIon
IS an adequate. .approxlmatlon;
(c) PI of the well; and (d) stabIlIzed
production rate with 500-psi drawdown,

Referencesi
I I'L 1 a Ith ews,
Flo"'

C ..an
S
d R us~e,II D ..:
G
P re.\'sllrl' BId
III lip all d
Tests in Wells, Monograph Series, SPE, Dallas (1967) I.

2. van Everdingen,A.F. and Hurst, W.: "The Application of the
Laplace Transformation 10 Flow Problems in Reservoirs,"
T~an,\'.,AIME(1949) .186,305-324.
...
Slider,
H.C.:
PractIcal
Pt!troleliln
ReservoIr
Engmt!t!rmg
Methods, Petroleum Publi~hing Co., Tulsa (1976) 70.

4. Hawkins, M.F. Jr.: "A Nole on the Skin Efrect," Trans.,
AIME (1956)207,356-357.
5. Odeh, A.S.: "Pseudosteady-Slale Flow Equation and
Productivity Index for a Well With Noncircular Drainage
J. Pet. Tech.(Nov. 1978)1630-1632.
(,. A~arw;1I, R.G., AI-IIII~~;1iIlY,R., ;1l1dR;1mcy,II.J. .Ir.: "All
Invc~tigationof Wellhorc Storageand Skin Efrcct in Un~teady
Liquid Flow -I. Analytical Treatment,"
(Scpt. 1970)279-290; Tran,\'., AIME, 249.

Soc. Pt!t. Eng. J.

7. Wattenbarger,R.A.andRamey,H.J.Jr.:"Anlnvestigation
of Wellhore Storage and Skin Erfect in' Un~leady l.iqllid
Flow-II. Finite-Dirference Trcatment," Soc. Pt!t. Eng. J.
(Scpt.1970)291-297;Tran.\'.,AIME,249.
R. Kat7, D. L. f't 01.: lIandhook of Natural Gas Engineering,
Mc(iraw-llilllJfIOkCo.lnc.,NcwYork(1959)411.
9. (ar~l;1w, II.S. and Jaegcr, J.C.: Conduction of "('Ot in
,lilllill\', ~ccondcd., Oxford althe ClarendonPress(1959)25R.
10. Earlougher, R.C. Jr.: Advances in Wt!1I Test Analysis,
Monograph Serie~,SPE, Dallas (1977)5.
II. Matthews, ~.S". Brons, F., and Hazebroek,~.: "A Method
Determination of Average Pressure In a Bounded
Re~ervoir," Trans.,AIME(1954)20I,182-191.
12. Horner, D.R.: "Pressure Build-Up in Wells," Proc., Third
World Pet. Cong., The Hague(1951)Sec.11,503-523.

.Chapter

2

,

Pressure Buildup Tests
2.1 Introduction
This chapter discusses the most frequently used
pressure transient test, the pressure buildup test.
Basically, the test is conducted by producing a well at
constant rate for some time, shutting the well in
i (usually at the surface), allowing the pressure to build
up in the wellbore, and recording the pres~ure
': (usually downhole) in the well bore as a function of
! time. From these data, it is frequently possible to
estimate formation
permeability
and current
drainage-area pressure, and to characterize damage
or stimulation and reservoir heterogeneitie~ or
boundaries.
The analysis method discussed in this chapter is
based largely on a plotting procedure suggc~ted by
Horner.1 While this procedure is strictly correl.'t only
for infinite-acting reservoirs, these plots also can be
interpreted correctly for finite reservoirs,2 so only
this plotting method is emphasized. Another important analysis technique for buildup t~~t~, u~ing
type curves, is discussed in Chap. 4.
The chapter begins with a derivation of the Horner
plotting technique and the equation for calculating
skin factor. Differences in actual and idealized test
behavior then are discussed, followed by comments
on dealing with deviations from assumptions made in

test in an infinite, homogeneous, isotropic reservoir
containing a slightly compressible, single-phase Iluid
with constant fluid properties. Any well bore damage
or stimulation is considered to be concentrated in a
skin of zero thickness at the wellbore; at the instant
of ~hut-in, flow into the well bore I.'easestotally. No
actual buildup test is modeled exactly by this
idealized description, but the analysis methods
developed for this case prove useful for more realistic
situations if we recognize the effect of deviation from
~ome of these a~~umptions on actual test behavior.
Assume that (I) a well is prod~lcing from an infinite-acting re~ervoir (one in which no boundary
effects are felt during the entire flow and later shut-in
period), (2) the formation and fluids have uniform
properties, so that the Ei function (and, thus, its
logarithmic approximation) applies, and (3) that
Horner's pseudoproducing time approximation is
applil.'able. I f the well ha~ produced for a tim~ I p at
rate q before shut-in, and if we call time elapsed since
shut-in ~I, then, using superposition (Fig. 2.1), we
find that following shut-in
8
I 688
2
Pi-Pws=
-70.6~[lnl-!--~!-!:!]-'-2sJ
kh
k (Ill + ~/)

developing the Horner plotting technique. We then
examine qualitatively the behavior of actual tests in
,i common
situations.analysis
The procedure
chapter next
develops inreservoir
detail a systematic
for

-70.6

( -q) 8
I 688 cI>
kh 1J./ln( ~---~)k~1

buildup tests: (I) effects and duration of afterflow
(continued production into the well bore following
sllrface
shut-in),and
(2)stimulation,
determination
of permeability,
(3) well damage
(4) detc.:rnlination
of

which becomes

pressure level in the surrounding formatioll, and (5)
reservoir limits tests.
Up to this point, the analysis procedure discussed
is applicable only to single-phase flow of a slightly
compressible liquid. The chapter concludes with a
discussion of how the procedure can be modified to
analyze tests in gas wells and in wells with two or
three phases flowing simultaneously.

or

.q81J.

1.1 The Ideal Buildup Test
I n t h IS section we denve an equation d escn" b'Ing an

..

..

ideal pressure buildup test. By ideal test we mean a

8
PlY.\"
=Pi -70.6~ kit

2
-2 S] '

lnf (I II + ~/) I ~/l,

8
Pw~'=Pi -162.6!!~

log! (Ip + ~I) I ~/]. ...(2.1)
kh
The form of Eq. 2.1 suggests that shut-in BHP, p,
recorded during a pressure buildup test should plotW~s
a straight-line function of log [(I +~/)/~/].
Further, the slope m of this straight line fhould be
m= -162.6-

.
kh

It is convenient to use the absolute value of In i~ test

t"
~
i
!'

ri{-

'

WELL TESTING

t
Q.

""'"'"

p

Pi

"'"
ws

m

w

~ ~tp

..~l\l

a::

0

~t

1000

TIME

use the
number

Fig. 2.2-Plotting technique for pressurebuildup test.

-I

s= 1.151

( ~~

that
(2.2)

kh

formation permeability, k, can be determined
a buildup test by measuring the slope In. In
if we extrapolat:e this straight line to infinite
illlit-in time [i.e., (II!+~/)/~/=I)
the pressure at
:hislime will be the original formation pressurePi'
Conventional practice in the industry is to plot PII'.~
(I" +il/)/~1
(Fig. 2.2) on semilogarithmic paper
values of (/p+~/)/~1
decreasing from left to
The slope "1 on such a plot is found by simply
the pressures at any two points on thc
i!raight line tha~ are one cycle (i.e., a factor of 10)
Ipart on the semllog paper.
We also can dctermine skin factor s from the data
in the idealized pressure buildup test. At the.
Ilslant a well is shut in, the nowing BliP, Pwj' i~
.2
ln 1,688 </>Jl.(('"') -2S
P 11./'
'=P.+70.6~
kll
kl"
/

+1.15110g

688
'

2:onvention
</JJl.C('W

)and

kill

( !2~
)
Iprhus,

(2.3)

It is conventional practice in the petroleum industryfrom
to choose a fixed shut-in time, ~/, of I hour and the!dditio
corresponding shut-in pressure, PI hr' to use in this
equal ion (although al'Y shut-in time and the
corresponding pressure would work just as well). The
pressure, PI hr' must be on the straight line or its(S.
extrapolation. We usually can assume further thatNith
log (/(,+~/)/lp
is negligible. With these sim-'ight.
plificatlons,iubtracting
-k
S=I.151
(Plhr-Pwj) -Iog(
_2)+3.23
In
</JJl.C('
w
(2.4)Ivailab
"
Not.e, again that. the .slope '~1 IS consIdered t~ be a
pO.~ltlvenumber In thIs equation.
In summary, from the idcal buildup test, we can
determine formation permeability (from the slope m

].

[

of the plotted te~t re~ults), original reservoir
pre~sure,Pi' and skin factor, s, which is a measure of
damage or stimulation.

B
=Pi + 162.6;{-

2
.[log(!~~c:h)-0.869S]
kip
1 688 </JC ,2
=Pi +I"llog(~-~)
-0.869sJ.
kIp
It ~hut-in time ~I in the buildup test,
-viscosity,
PII..~-Pi-I,'log[(/p+~/)/t1/].
these equations and solving for the skin
S, we have

--!~,

) + 1.15110g(

In

l (

I

~t

Fig. 2.1 -Rate history for ideal pressurebuildup test.

-~
111-162.6

10

tp + ~t

=0

analysis; accordingly,
in this text we will
that ," is considered a positive

100

Example 2.1-Analysis

of Ideal

Pressure Buildup Test
Problem. A new oil well produced 500 STB/D for 3
day~; it then was shut in for a pressure buildup test,
during which the data in Table 2.1 were recorded.
For this well, net sand thickness, h, is 22 ft; formation volume factor, Bo, is 1.3 RB/STB; porosity,
</J,is 0.2; total compressibility, c" is 20x 10-6; oil
Jl.o' is 1.0 cp; and well bore radius, , w' is 0.3
ft. From these data, estimate formation per-~omhin
meability, k, initial reservoir pressure, Pi' and skinactor
factor, s.

';-~t'.

-

PRESSURE BUILDUP TESTS

23

TABLE 2.1 -IDEAL PRESSURE
BUILDUP DATA

Time After
Shut.ln At
(hours)

PWI
(pslg)

0
2
4
8
16
24
48

1,150
1,194
1,823
1,850
1,876
1,890
1,910

TABLE 2.2 -BUILDUP TEST DATA
FOR HORNER PLOT

At
(hours)
2

~

4
8
16
24
48

ZOO()

P"

PWI
(pslg)
1,194

19.0
10.0
5.5
4.0
2.5

1,823
1,850
1,876
1,890
1,910

EARLY
TIMES

t

I

~t
31.0

MIDDLE
TIMES

LATE
TIMES

P ws
'

.'"

01

,---

Vi

a.
VJ

~

"
""

100

<-P,.-

~ «>

I

I

og

tp + At

l1t -

~t

.-

Fig. 2.3 -Ideal pressure buildup test graph.

Fig. 2.4-Actual buildup test graph.

Solution. To estimate permeability, original reservoir
pressure,and skin factor, we must plot shut-in BHP,

I
-og

P ,vs. log (I + AI) / AI; measure the slope /11and
u;: Eq. 2.2 toPcalculate formation permeability, k;

= 1.43.

extrapolate the curve to [( I P + AI) / AI] = I and read
original reservoir pressure, Pi; and use Eq. 2.4 to
calculatethe skin factor s. (See Fig. 2.3.)
Producing time, I , is given to b~ 3 ~ays., or 72
hours (in this case, Horner's approXimation ISe~-acI
bccau~ethe well was produced at con~tant rate since
timezero). Thus, we develop Table 2.2.
.of
We plot these data, and they fall along a straight
line suggested by ideal theory. The slope /11 of the
straight line is 1,950 -1,850 = 100 psi (units are
actuallypsi/cycle). The formation permeability k is
q81J. (162.6)(500)(1.3)( 1.0)
k = 162.6
=
=48md.
fllh
(100)(22)
From extrapolation
of the buildup curve to
1(/p+~/)/A/)=I,Pi=I,950psig.Theskinfactors
isfound from Eq. 2.4:
[ (Plhr -Pwj)
( k
)
]
s=I.151
-log
2 +3.23.
m
<P1lC
/' w
Thevalue for P ws is PI hr on the ideal straight line at
(/p+A/~/A/=(72+
1)/1 =73; this value is PI hr =
1,764pslg. Thus,
[ (1,764-1,150)
s= 1.151
(100)

( (0.2)(1.0)(2 48
) 3 23
x 10-5)(0.3)2 + . 1

This means the well has a now restriction.
.
2.3 Actual Buildup Tests
Encouraged by the simplicity and ease of application
of the ideal buildup test theory, we may test an actual
well and obtain a mo~t di~collraging re~ult: Instead
a single straight line for all times, we obtain a
curve with a complicated shape. To explain what
went wrong, the radius-of-investigation concept is
useful. Based on this concept, we logically can divide
a bllildllp curve into three regions (Fig. 2.4): (I) an
early-time region duri.ng wh~l:ha pressure transient is
Illoving tllrough the lormatlon n\:ar\:~t tile wcllbur\:;
(2) a middle-time region during which the pressure
transient has moved away from the wellbore and into
the bulk formation; and (3) a late-time region, in
which the radius of investigation has reached the
wel.I's ~rainage b~undaries. Let us examine each
regIon m more detatl.
Early-Time Region
As we have noted, most wells have altered permeability near the well bore. Until the pressure
transient caused by shutting in the well for the
buildup test moves through this region of altered
~

.,'

-~permeability, there is no reason to expect a straight.j ,line slope that is related to formation permeability.
'( (We should note that the ideal buildup curve -i.e.,
'one
with a single
straight line
alldamage
time -t
is possible
for a damaged
wellover
onlyvirtually
when the
i~ concentrated in a very thin skin at the sand face.»)
There i~ another complication at earliest times in a
pressure buildup test. Continued movement of fluid
into a well bore (afterflow, a form of wellbore

""""'I'IU

Iq
W
~
~

l

~t

P

storage) following the usual surface shut-in compre~sesthe fluids (gas, oil, and water) in the wellbore.
Why should this affect the character of a buildup
curve at earliest times? Perhaps the clearest answer
lies in the observation that the idealized theory

L -- 0

J\
L1

leading
to
the equation
P"'.\,=Pj-'"
log
.(I" + ~I) / ~IJ explicHly assumed that, at ~I =0, flow.
rate abruptly changed from q to zero. In practice, q

.
Fig. 2.5-Rate history for actual pressurebuildup test.

declines toward zero but, at the instant of surface
shut-in, the downhole rate is, in fact, still q. (See Fig.
2.5.) Thu~, one of the assumptions we made in
deriving the buildup equation is violated in the actllal
te~t, and another question arises. Does afterflow ever
diminish to such an extent that data obtained in a
pre~sure buildup test can be analyzed as in the ideal
test? The answer is yes, fortunately, but the important problem of finding the point at which afterflow ceasesdistorting buildup data remains. This
is the point at which the early-time region usually
end~, becau~e afterflow frequently lasts longer than
the time required for a transient to move through the
altered zone near a well. We will deal with this

2.4 Deviations From Assumptions
in Ideal Test Theory
..,..
In suggest~ngthat tests 10glc~lly can .be divided Into
early-,. mIddle-, and late-tIme re~lons, we ha~e
recognl~ed that several. assu~ptlons made .In
developIng ~he theory of Ideal bulldu~ test ~ehavlor
are n,ot valid for actu.al t~sts.. In this section, we
~xam.me further. t.he Implications. of ~hree ove~Ideallze~ assumptlo~s: (I) the. In,finlte-reser~olr
assumptIon; (2) the single-phase. lIquid as~umptlon;
and (3) the homogeneous reservoir assumption.

problem. more ~omplctelY when we. ~iscu~s a
systematIc
analysIs procedure for pressure buIldup
te~t~.

.

TIME

~

Infinite Reservoir Assumption
'~I(t In developing the equation suggesting the Horner

,
...thc
Wh711the radll~~ of Illvc~tlgatlon ha~ moved hCYOlld
thc Illflucl1ce 01 thc altcrcd 7,one near thc tc~ted well,
and when arterflow has ceased di~torting the pre~~ure
buildup test data, we usually observc the idcal
straight lil1e ~hose slope is related to formation
permeability.' (This straight line ordinarily
will
conlilluc until tIle radius
of
il1vestigati0l1
rcachc~
OI1C
b
d
I
or more reservoir
OUI1 aries, massIve letero-

plot, we assumed that the reservoir was infinite
during
both the production period preceding
.'
.
buildup te~t alld the buildup test Itself.
l:rcqllcl1tly, the re~ervoir i~ at p~elldosteady-statc
hcforc ~hut-il1; if ~o, neither the Ei-function solution
nor its logarithmic approximation should be used to
describe the pressure drawdown caused by the
producing well:
B
(p.I -p \I f ) pro d "c II ~ -70.6~~
kl1"

gencities, or a fluid/fluid contact.)
Sy~tematic analysis or a pressure buildup test using
the Horner method of plotting
IIl\l'S vs. log
<,f
I' + ,~I) / ~f] requ!res th~t we recognize thi~ middl~tllnc llnc
tlInt, Inlil1cs
partin
Icular,
wc do al1d
not con
fu~c It
with
falseal1d
straigllt
the carlylate-til1lc

2
.f In [ 1,688 c/>ILc,r"'
1 -2s1.
{k(l"
+ ~I)
j
II1~tead, if the well is centered in a cylindrical
rc~crvoir '

.

III
-.acting
M IC
C e-' I '.Ime ,(,~ICt"

.

..

regions. As we have seen, detcrmination of reservoir
pcrmeability and skin factor depends on recognition

(p. -p)
,..f

or the middle-time line; estimation of average
drail1age-area pressure ror a well in a developed field
al~o requires that thi~ line be derined.
1.l1tl'-llml' Itl'~lctn
Given enough time, the radiu~ or invc~tigation
eVcl\tl;ially will reach the drainage boundaries of a
wrIt. (In this late-time region pre~~ure bchavior is
innllcnccd hy boul1dary col1figuration, il1tcrfcrcl1cc
from nearby wel15, signiricant reservoir
gcncities, al1d nuid/fluid contacts.)

~..~

I

hetero-

= 141 2~
prodwell.
kh

. 1 0.000527 k (I p + ~I) + In
c/>/lc,r; ---rlt.

( ~ ) -~4.]

Thus, we must conclude that in principle, the Horner
plot is incorrect whcll the reservoir is not infinite
acling during the flow pcriod preceding the buildup
, test. Boundaries become important as rj -r f'.
The problem is compounded when rj -r (' arter
shut-in. Then, too, the Horner plot is incorrect in
pril1ciple.
This difficulty

";;,,,;;

-

is resolved in different

-

ways by

.I

PRESSURE BUILDUP TESTS

25

MTR

I t

(I a 2)

t

Pws

~ I

Pws

-

ETR-1M'TiR

I-!:~f (I)
Pwf(2)

Pwf(182)

tp + At
-+--

Fig.

2.6 -Buildup

test

wellbore

different

will

by

use

the

reservoir

I,

This

will

Cobb

for

all

without

plots;

of

plotting

a

method

when
during

test)

for

Single-Phase
We

The

the

only

the

reservoirs

the

mobile

For

plot
At-~

offers
not
al

the

this

test

of

the

than

from

for

reservoirs

curve
will
'j
during

convenient
found
in

shut-in

a

al

plot

plotting

means
of
some
other

lime

is

a

in

only

which

useful

Even

at

even

from

of

greater

method

shut-in

developed

deviate
from
reaches

"w'

are

times

Co

=

at

the
ideal
reservoir

8

--!!-

+

plotting

than

t

'

es

.

b
M II
y
I er,
Slider.4
Many

method

simpler
t

d

be

Horner

' ddl

t

a ml

e-

by

-2,
80

'

d

.

tchl

region

'

escfl

. nson

the

3

data-

because

Consider

discuss

e

it

for
is

a buildup

b d b

y

evaluation

of

compressibility,

,

(2.5)

dp

~

~
dp

+

~
8

Eq

2

(2
dp.'

,

.

.W

u

MDH

method.

Ime

H
use

~O,

wat~r

(1)

flow

of

modIfications
,
single-phase
~wo

or

correlatIons

1 .and

to

three

~seful

other

fluId

.

the

gas

flow

flow

equatIons

and

phases.

~2)

Ap~endlx

.
.1..1;

us to mou'-'.
h.
l IS C I 1arter

...

required
.
slmultane.ous

D sum~~r!~es

for.calculatm~

propertIes

6)

':,.'

"
'b '
l
I
h
II
ese compressl
Iity
re at Ions
IpS a ow
I
I
fl
f
'I
L
..
,
sl.ng
e-p 1ase,
ow.o
01,
ater
sections
I~

iden-

reservoirs

and
yes,
analysts

suggested

the

th

WI

D

S

(1.4)

dR

~

dp

-~
C II' -8.

'

d '
are
Iscusse
(MDH)
and

total

equations:

complicated:

Th

can

ignored.
use

.,..,..,...

when
and

we

cases,

The

slope
bef?re
boundaries.

However,
the
middle-time
region
still
t f' d
t f
I
I
t '.
I Ie ,excep
or
ong
ear y- Ime
regIons,
Other
analysis
methods
for
finite-acting

be
if

flow

+cf'

flow,

d8

80

shut-in.2

lo

iman

many

cannot

solutions

"0'

somewhat
-I

in

an
contain

account

+CgSg

in single-phase

specifically

state

+cwSw

compressibility,

Also,

into

Even

contain
also

formation

contains

modified.

many

the

in

be

flows

taken

Ct'

oil

the

oil

saturation.

are

reservoir

must

saturation;

factors

(1) williout

Iluw.

petroleum

liquid

CI =coSo

per-

accurately

pseudosteady

begin
to
shut-in

These

a

single-phase

gas

damage:

altcr

Assumption
that

water

formation

(2) willi

l.iquid

assumption

~f

with

and

compressibility

which,

formation

determined

Horner

at

a

reservoirs,

be

a

for

analyst.

finite-acting
can

theoretically

one

test

compressibility,

pressure
for

correct

(i,e.,

e).

to

meability
slope

<r

Horner

the

is

reservoir

r;

checkpoint

Buildup

immobile

method

2, The
extrapolating

2.1-

log

alterllow

Smilh,2

(even

the

Fig.

dafllage.

use

and

tests

preceding

infinite-acting

tp.. + At.

3.

we

of

(1)

wellbore

reasons.

an

time

alterllow:

pseudosteady-state

period

following

-+--

(2) witll

text,

plot

reached

production

for

this

research

Horner

has

no

arid

In

the

tp + At

-15

with

damage

analysts,

supported

,

log

compre.sslbliltles

needed

In

analysIs

of

well

...

tests.
p ws = p;

-m

log

(t

+

AI)

/ AI

p
=p;-mlog
:

If

t p )10AI

\

p

(tp+At)+mlogAt.

during

f examined,

Ilomogcneou~

the

then

log

= constant

range

of

+A/)

=log

(/p

+ m

log

This

time

values

tp = constant,

and

most

real

p

to

the

ys.

log

;iot

(in

Further

insight

provided

by

Exercise

plotting
At.
the

technique

It

has

time

into

this
2.2.

the

suggested

same

range
plotting

of

slope

m

by
as

the

applicability).,
technique

is

average

solutions
include

particularly

fluid

are
portion
to

flow

changes

only
prove

solutions
for

to

to

the.

homog~llcous
adequate

be

eariy

in

time

properties.

encountered
of

the

equations
of

depositional

When
(particularly

reservoir),
lose

accuracy.

while:

environment,

test
by
massive
in

the

:
;

for

the
tested
well
dominate
pressure
change
is dominated

and

heterogeneities
localized

is

yet

are
valid
~olutions

nearest
Rate
of
rock

A~~"mption

homogeneous,

reservoirs,

coQditions
.behavior,

leads

Horner

shut-in

reservoir

now
eqilatim1S
re~ervoir~,
The

At

ws

MDH:

No

Re~ervoir

simple
Examples
with

i
!
I

a

-MTRILTR-

t

E

R

p!,

Pws

w
-4-

-+-

Fig. 2.8 -Buildup

log

tp + L\t
._-~t

test in hydraulically

0-40-

fractured

well,

resultant changes in permeability or thickness, and
some nuid/nuid contacts. The longer a test is run,
the higher the probability
that a significant
het.eroge~eitr will be enc?mpassed within the radius
of InvestIgatIon and thus Influence the test.
Modifications to the simple reservoir models have
hccn developed for some important reservoir
IIctcrogcncitic~. Still,
in actllal lIeterogcncou~
rc~crvoir~, tile te~t analy~t mu~1 be aware con~talltly
of tllc po~~ibility of an unknown or impropcrly
modelcd IIctcrogcncity. Thc~e IIcterogcncitics makc
analysis of late-time data in tran~icnt test~ more
difficult -reservoirs are rarely uniform cylinders or
I'arallclcpipcds, and thc analy~is technique that is
I'a~cd on tllc~c a~~uml'tion~ for Ircatlncnt of laletimc data can bc difficult to apply witllOllt ambiguity.
What is the test analy~t to do with late-time data7
Opini{~n~ vary: One frequent approa~h i~ t? use
anaIY~I~ tccllnlque~ ~uggested by published simple
m{,dcl~ -but to try to find other models tJlat also fit
the observed data. One then chooses the most
probable reservoir description, and recognizes that
thc analysis may be ah.~o/"tel_I'"'correct.
Q I..
I) I
.
r L" Id "
.faflve Je lavlor 0 ..Ie
I e~f~
2...lIa
S
We IIOWhave devcloped the background requircd to
under~tand the qualitative bellavior of commonly
occurring pressure buildup curves. There is an important reason for this examination of behavior. It
provides a convenient means of introducing some
factor~ that innuence the~e curve~ and that can
(,h~cure il,lcrl,rclation unlc~~ Ihey are rccognized. In
thc figllre~ that follow, the carly-, middlc-, and latetime regions are dcsignated by ETR, MTR, and LTR,.
rc~pcctivcly. In tl,ese curves, the most important
region is the MTR. Interpretation of the te~t using
the ll{,rncr plot I" \I',tvs. log (f l' + d/) I d/] i~ usually
iI'lpos~ible unlc~s the MTR can be recognized.

L TR -+

tp + At

log/-At

Fig. 2.9 -Boundary
effects in pressure buildup test: (1) well
centered in drainage area and (2) well off.center
in drainage area,

Fig. 2.6 illustrates the ideal buildup test, in which
the MTR spans almost the entire range of the plotted
data. Such a curve is possible for an undamaged well
(Curve I, with the level of Pwi' the nowing pressure
at shut-in, is shown for reference) and for a damaged
well with an altered zone concentrated at the
wellbore. This latter situation, shown in Curve 2, is
tnuicated by a rapid ri~e in pressure from nowing
pre~~ure at sllut-in to tile pressures along the MTR.
Ncitllcr ca~c i~ ob~crved often in practice with a
~urface ~hut-in bccau~c afternow usually distorts the
early data that would fall on the straight line.
Fig. 2.7 illustrates the pressure buildup test obrained for a damaged well. Curve I would be obtained with a shut-in near the perforations
(minimizing tile duration of afternow); Curve 2
would be obtained with the more conventional
surface shut-in~' Note in this figure that the nowing
BHP at shut-in, PIII(' is the same for either case, but
that the afternow that appears with the ~urface shutin (I) completely obscures information rcnecting
near-well conditions in the ETR and (2) delays the
beginning of the MTR) A further complication introduced by afterno~ is that several apparent
straight lines appear on the buildup curve. The
qllestion ari~e~, how do we find Ihe straight line (the
MTR line) whose ~Iope is related to formation
permeability7 We will deal with this question shortly.
Fig. 2.8 shows characteristic behavior of a buildup
test for a fractured well without afternow. For such a
well, the pressure builds up slowly at first; the MTR
develops only when the pressure transient has f'\1oved
beyond the region innuenced by the fractureliin a
buildup test for a fractured well, there is a possioility
that boundary effects will appear before the .ETR has
ended (i.e., that there will be no MTR at all).)
Fig. 2.9 illustrates two different types of behavior
in the LTR of a buildup test plot. Curve I illllstrates
middle- and late-time behavior for a well reasonably,

j
~

~

-j

buildup te~t alone i~ not ~uffil:ient to ilJdil:atl.' IIII.'
presence or absence of afterllow -it is merely a clue
that sometimes indicates presence of afterflow.
A log-log graph of pressure change, p",s -Pili'
in a
buildup test vs. shut-in time, L\t, is an even more
diagnostic indicator of the end of afterllow
distortion. Fig. 1.6, based on solution~ to the Ilow
equal ions for I:onstant-rate produl:tion with wellborl.'
sl(lragl.' cJislorliolJ. d(."scrib(."sprl.'SSllr~ builcJllp Il.'sls,
as Wl: Ji~~u~!i in !iom(."Jl:tail in <"Ilap. 4. I.'or u~(."()f
thi~ figure for buildup test~, dimen~ionle~~ pre~~url.',
PD' is defined as

t
Pws

PD =
log

tp!

At
At

Fig, 2.10-Characteristic influence of afterflow on Horner
graph.

0.00708 kh(pws -P"'f)

'.

(2.7)

q8J1.

Dimen~ionless time, I v' and dimen~ionless wellbore
storage constant, CsD' are defined e~sentially as for
con~tant-rate production:
0.000264 kAle
IV=
2'
(2.8)
<pJl.c,r w

centered in its drainage area; Curve 2 illustrates
behavior for a well highly off center in its drainage
area. For simplicity, the ETR is not shown in either
case.
Many curve shapes other than those discussed
above appear in practice, of course. Still, these few
examples illustrate the need for a systematic analysis
procedure that allows us to determine the end of the
ETR (usually, the time at which after flow ceases
distorting the test data) and the beginning of the
LTR.

0 894 C,
CsD = _':' ;f,
<PJl.c,w
where
A
Cs =25.65 ~
P
for a well with a rising liquid/gas
wellbore, and
C -,
!)-(tt-lJ V wb

Without this procedure, there is a high probability
of choosing the incorrect straight-line segment and
using it to estimate permeability and skin factor.

for a w~llbore containing
(liquid or gav.
We define

"

(2.9)

interface in the

only single-phase fluid

2.6 Effects and Duration of Afterflow
Ale = AI/( I + AI/I p).
(2.10)
In our discussion thus far, we have noted several
problems that after flow causes the buildup test
As noted in Chap. I, wellbore storage distortion
analyst. Summarizing, these problems include (I)
(afterflow in the case of a buildup test) has ceased
delay in the beginning of the. MTR, making its
when the graphed solutions for finite CsD become
recognition more difficult;
(2) total lack of
identical to those for CsD = O:~/Also,a line wiOJ unit
development of the MTR in some cases, with
slope (450 line) appears at earl~ times for most values
relatively long periods of afterflow and relatively
of CsD and s. The meaning of this line in a buildup
early onset of boundary effects; and (3) development
test is that the rate of afterflow is identical to the flow
of se~eral false straight lines, anyone of whil:h could
.r~te just before shut-in. ).
be mistaken for the MTR line. We note further that
A,lf the un1t-slope line is present, the end of afrecognition of the middle-time line is essential for
terflow distortion occurs at approximately one and a
successful buildup curve analysis based on the
half log cycles after the disappearance of the unitHorner plotting method Ipws vs. log 1(/p+A/)
slope line. Regardless of whether the unit-slope line is
/ AI JI. becau~ethe line mU!it b~ identified to e!itimate
prc~clll, thc end of aftcrllow di!itortion can be
reservoir permeability, to calculate skin factor, and
determined by overlaying the log-log plot of the test
to estimate static drainage-area pressure. The need.
data onto the Ramey solution (Fig. 1.6) -plotted on
for methods to determine when (if ever) afterflow .graph or tracing paper with a scale identical in size to
ceaseddistorting a buildup test is clear; this section
the Ramey graph -finding any preplotted curve that
fills that need.
matches the test data, and noting when the preplotted
The characteristic influence of afterflow on a
curve for finite value of Cosvbecomes identical to the
pressure buildup test plot is a lazy S-shape at early
curve for CsD =0. This point, on the actual data
times, as shown in Fig. 2.10. In some tests, parts of
plot, is the end of afterflow or wellbore storage
theS-shape may be missing in the time range during
distortion)
which data have been recorded -e.g., data before
If the unit-slope line is present, we can use a
Time A may be missing, or data for times greater
relationship developed in Chap. I to establish the
than Time B may be absent. Thus, the shape of the
value of CsD that characterizes the actual test. There,

IY~L.L.

,
Ii.

I t:~

I IN\,)

,.,

TABLE2.3-OILWELL PRESSURE
BUILDUPTESTDATA
AI

I +AI

L-

-(~~~~~~-- ~0

~t.=~1/(1+~)

--3,534--

Ip

Pws

~~ours)

(psla)

Pws -Pwf

(psla)

0.15

90,900

0.15

3,680

146

0.2
0.3
0.4
0.5

68,200
45,400
34,100
27,300

0.2
0.3
0.4
0.5

3,723
3,800
3,866
3,920

189
266
332
386

1

13,600

1

4,103

569

2

6,860

2

4,250

716

4
6

3,410
2,270

4
6

4,320
4,340

786
806

7
8
12
16
20
24

1,950
1,710
1,140
853
683
569

7
8
12
16
20
24

4,344
4,350
4,364
4,373
4,379
4,384

810
816
830
839
845
850

60

228

59.7

4,405

871

72

190

71.6

4,407

873

30
40
50

455
342
274

29.9
39.9
49.8

4,393
4,398
4,402

-

.

859
864
868

/

we noted that any point on the unit-slope line must
satisfy the relationship
CsoP 0

= I,

,

0

".443141.4431+tf7a

(1.42)

to
which, in terms of variables with dimensions, leads to
Cs=

qB At
24~'

(2.11)

.~
~
0.:

-

where ~t and At] are vailles read from a point on the
lIllit-slopc line. If we can estahlish C.t in this way (a
less acceptable alternative is to use the actllal
mechanical properties of the well- e.g., Ct = 25.65
A\I'/I/PI,.h for a well with a rising liquid/gas interface), we then can establish CsO from EQ. 2.9 and
thus determine the proper curve on Fig. 1.6 on which
to attempt a curve match. (It is difficult to interpolate between values of CsO on this curve; accordingly, many test analysts prefer to find a match
with the preplotted value of Cso closest in value to
the calculated value.) With Cso established, and
permeability, k, and skin factor, s, determined from
complete analysis of the test, we can use the empirical relationships below to verify the time, tl,'h.t'
marking the end of well bore storage distortion.
to=50CsoeO.14S,
or

170 (XX)C
, ",/I.t=

(2.12)
O.14,t

se.
(2.13)
(kh/lJ.)
We will illustrate (I) the application of the basic
curve-matching procedure in Example 2.2; (2) the
check provided by Eq. 2.13 in Example 2.4; and (3)
COillplctc, quantitative curve-matching procedurcs in

Chap. 4.

'

In thi~ di~clls~ion of the qualitative application of
curve matching, we al~o should note that appearance
of boundary effects or the effects of heterogeneiti'es
frequently can be verified from the curves. Fig. 1.6 is

~

~

Id'.,.

r/
lpt~t
~t

Fig. 2.11 -Semi log graph of examplebuildup test data.
a s.olution to the flow equ~tions for .an iijfiniteactIng, .homogeneous res~rvolr; when, In an actual
reservoIr, a pressure transIent reaches a boundary or
important heterogeneity, the actual test data plot will
deviate from Fig. 1.6. This characteristic of curve
matching is illustrated in Example 2.2.

Exal11pte2.2 Finding the End
of Wet/bore

Storage Distortion

Pr()blem.ThedatainTable2.3wereoblainedina
pressure buildup test on an oil well producing above
the bubble point.
The well was produced for an effective time of 13,630
hours at the final rate (i.e.. , p = 13,630 hours). Other
data include the following.
qo =
1J.0=
<I>=
B =

250 STB/D,
0.8 cp,
0.039,
1.136 RB/STB,

f
t.

l

-

PRESSURE BUilDUP

-~'"
TESTS

",
'w

= 17 x 10 -6
= 0.198 ft,

'e

=

29

psi -I,

Then

1,489 ft (well is centered in square
drainage
area, 2,640 x 2,640 ft; r
f
I
.e
d.
ra IUS 0 clrc e with same area),
53 Ibm/cu
ft,

C
is

.

Po

=

,

Cs
hr2
,
w
(0.894)(0.0118)

= (0.039)(1.7x

rising liquid

level in well during

A semilog

graph

isvs.shown
Ale)

Eq ..,2 9

-0.894
sD -/f>c

Awb=0.0218sqft,
h = 69 ft, and

(Pws
these -Pwf
data

from

[Pws

vs.

shut-in.

log

Thu~,

(/p+A/)/A/]

of

in Fig. 2.12.
2.11, and
Froma log-log
these graph~,
graph

answer the following
questions.
.I. ~t what shut-in time (AI) does

afterflow

cease

matchinl

10-S)(69)(0.198)2

~hould

be attempted

-=5880I
'

.

in Ihe

rangc

103<CsD<10.
2. 71)etermlnatlon

of

Pernleability

In chis scction,
we examine
techniques
for Ihe nexi
step in the sy~tematic
analysis
of a pre~sure buildup

dIstorting the pre~~ure buildup
tc~t data?
2. At what shut-in time (AI) do boundary
effects
appear?
Solution.
From
the semilog
graph
(Fig.
2.11),
it
seems plausible
that afterflow
distortion
disappears
at (I p + AI) I A/.= 2,270 or AI = 6 hours because of t he
end of
the
characteristic
lazy-S-shaped
curve.
However,
other reservoir
features
can lead to this
same shape, so we confirm
the result with the log-log
graph.
After
plotting
Ap = P
-P
vs
Ale = AII(I + Alii)
on log-log paperwswe fi~d tha~
the actual data m well. curve~ for s::: 5 for scveral
valuesofCsD(e.g.,CsD=103,104,andIOs).lneach
case, the curve fitting
the earliest data coincides
with

or
falloff
test:
determining
bulk-formation
pcrmeability.
llecau~e
bulk-formation
pcrmeability
i~
obtained
from the slope of the MTR
line, correct
select!~n
of this
region
is critic.al.
Average
permeablhty,
kJ'
also
can be estimated
from
informati~n
available
in.bu.ildu~
!.ests:
~he II~st problcm
IS I~entlllc~tlon
of Ihe MTR.
This
region
cannol
begm
until
aflerflow
cease~
distorting
the data; indeed,
cessation
of afterflow
effects
usually determines
the beginning
of the MTR.
If the altered
zone is unusually
deep (as with a
hydraulic
fracture),
passage of the transient
through
Ihe region. of Ihe ~rainage
a~ca.influenced
by Ihe
fracture
will determine
the beginning
of the MTR.

the CsD =0 curve for s= 5 at Ale =A/=4
to 6 hours.
This, then, is 1he end of wellbore
effects:
I
.= 6
hours. The data begin to deviate from the ~g:nilog
straight line at (t + AI) I A/.= 274 or At = 50 hours.
On. the log-log
gr~ph, data begin falling
below the
filling curve at At = At .= 40 hours
consistent
with
the semilog graph.
e'

.P~edicting
the ti!11~ at which. the ~TR
en~s is more
difficult
than predIcting
when It begIns. Basically,
the
mid~le-!ime
li~e
ends
when .the
radius
o!
investlgatlon
begins to detect drainage
boundarIes
of
Ihe leste~ well;
at this time,
the .pressure b~ildup
curve begins to bend. The problem
IS that the time at
which
the middle
region ends depends
on (I) the

In summary,
basing our quantitative
judgment
on
the more sensitive
semilog
graph,
we say that the
MTR spans the time range of AI = 6 hours to AI:; 50
hours. This judgment
is verified
qualitatively
by the
log-log
graph
curve
matching.
Even though
the
semilog graph is more sensitive (i.e., can be read with
greater
accuracy),
it alone
is not
sufficient
to
delermine
the beginning
and end of the MTR:
matching
Ramey's
solution
is a critically
important
part of the analysis.
The
log-log
curve-matching
analysis
was performed without
knowledge
of CsD' Note that CsD
can be established
in this
case,
at least
approximately:
from the curve match, we note that the

di~tance.
from
the te~ted
well
to the re~ervoir
boundarIes,
(2) the geometry
of the area drained
by
the well, and (3) the duration
of the flow ~eriod as
well as the shut-in period.
Cobb and Smith
present
c.harts that allow the analyst
to predict
the shut-in
time At at which the MTR should end if drainagearea geometry
and producing
time are known.
If this
i~l:ormation
is available
and if .the reservoir
i~sufficiently
~omoge~eous
that, untIl. the L TR begIns, it
behaves
In the Ideal way requIred
by Cobb and
Smith's
theory,
their charts
can be used to check
results.
...
One useful generalizatIon
can be made from their
result~.
If a well was at pseudosteady-state
before

data are
Ramey's

shut-in~
the
approximately

near the
solution;

unit-slope
the point

line on the graph
Ap = 100, AI = 0.1

of
is

essentially on this line. Thus, from Eq. 2.11,
qB AI
(250)( 1.136) (0.1)
Cs'= --=
-wcll
24 Ap
24
(100)
= 0.0118 bbl/psi.
..times
Alternatively
(and, m general,less
accurately),
C -25.65
A wb -(25.65)(0.0218)
s -p
-53
=00106
bbl/
.The
.pSI.
'Data are plotted on 3 x 5 cycle log. log graph paper (11 x 16'1. in.) and matched
w,~~~~~ Ramey solu,tion (such as provided In the SPE type-curve package)
p
n the same size scale.

time AI at which the L TR begins.
is
A/j" := 38 /f>J!cIA I k for a well cenlered

in a square or circular drainage area. In Ihe equation,
A (sq ft) is the drainage area of Ihe teste~ well. I f the
was not at pseudosleady-stale,
A/f/lS larger than
calculated
by Ihe rule above.
In many cases, we
~imply a~sume thai Ihe straight
line spanning
the
between Ihe end of after flow distorlion
and a
later bend of the Horner
plot constitutes
the MTR.
Use of the log-log
graph and curve matching,
as in
Example 2.2, can help confirm
this assumption.
calculated
radius of investigation
(r;) at the
assumed
end of the MTR
provides
a qualilative
'Cholce of time at which l TR begins Is somewhat arbitrary. The rule slated Is
based on a 10% deviation in slope ollhe Horner plot from the true MTR.

estimate

only

drainage

area

in the

In summary,

the

I

formation
1.
I

of

the

Determine

occurs

the

when

3.

infinite-acting

reservoir

sampled

given
for

determining

bulk-

by

probable

beginning

Horner

plot

roughly

the

of

the

becomes

a log-log
from
a curve
graph

948

permeability

lincar
MTR,
line,
and

at

4.

dt

=

50

establish
qualitative
5.

is no

its slope

the

assumed
but

MTR

or

can be used
be

permeability,

buildup

test.

in

MTR

it is so

1.19,

during

called

calculate

the

the

in

is valid

~

2
.q

B

II.

k J =

[I

/

n (r e r w)

only

a well

that

should

if

with

production

4

well,

kJ > k.

Average

consistency

the

for
analysis.

-pos~ibly

choice

MTR;

for

an

a

well,

in checking

If

k and

to

each

incorrect

method.

and

translating

near

u~uall~

-Estimating

operations:
fine

For

determIne

formatIon

Solution.

In

MTR

spans

ho\lr~

[2,270

note

that

III

the

= 4,437

tl~e

wIth
Wlt.h

can cause

time

2.2,
range

~ (t p + ~t)
slope

Example

2.2,

to

/ ~t ~ 274]

111 of this

-4,367

= 70

established

~t ~ 6 hours

straight

.From
line

that

the

to ~t ==50
Fig.

2.11,

is

jection

qBII.
lnh

method,
reasons

Wellbore

formation
through

damage

prop

p.ores

~eactlo~

the .drllllng

wIth
of the

fl~ld

(e.g.,

result!ng

fr~m

C<?mpletlon

reductIon

by

results

fluIds

as they

enter

delIberate

and

i~ di~~olving

in the

of

sand

hydraulic

plugging

open

some
when

acid

with

fluids,

usually

other
the

fracmalerial~

formation

special

or

at-

Common

near the wellbore
with
perforations.
Hydraulic

seams

fracture

from
productivity.

acidization

injection

the

agent

pressure

infrachighac-

that

will

creating

the

is removed.
2.4
the

shows
MTR

how
we can
is
identified

calculate
and

skin
factor,
bulk-formation

S,

is estimated:

[ (PI hr -Pwf)

(70)(69)

_I

(

m

k
,

)+

3 ..23]

I/>p.ct~

=7.65md.
It

i~

of

interest

is
is

completl?n
of

an~

formatIon

a well's

the

is creating

pressure

is
for

reduct~on

pl~gglng

permeabIlIty

it

permeability

or

~~trate).

usually

_ I 151
S.og.

(162.6)(250)(1.136)(0.8)

~he

I.ow-sallmty

improve

permeability

k=162.6-=

type-

..

companied

once

Then,

or

using

factor
physical

fluId

from

I~

include

fracture

.Eq.
psI/cycle.

deal

damage

drIllIng.

drillIng

Acidi7.alion

turing

we
of

easily

will

This.P,ermeability

filt!ate

s!mllar

the
In

We

f?rmatl.on.

techniques

permeabIlIty.

Example
the

..and
dlscu.s~ed

test

a more

when

In.cI~de

the

<?f clays

ttlring.

Pr()hle~.

is

calculating

method

of a well.

C:auses

form~tlon

skin
the

du!lng

m~terlal.ln

contact

Fonl1ation

when

(tp +~t)/dt

analyzing

applied

a wellbo~e.

occurs

StImulatIon
2.3

a
or

technique

vs.

it into

-a

in

damage

a well.

chapter

we examine
the
consider
briefly

term

other,

l'emleability

basic

md.

available

It involves

of

in a later

data

of Pws

of

method

stimulation

tempts
Example

a plot

drainage

is 7.65

estimate

The

factor

or

the

use

to

skin

descriptive

swellIng

analysis

well's

analysis.

k J do

of MTR.

to

test

from

another

reduced
k,

a stimulated

relationship

wrong

stimulated,

is valuable

test

proper

is

of

and

permeability

to incorrect

nor

permcability,

slope

buildup

the

something

the
k J < k;

in

bear

damaged

how

characterization

Before
useful
to

a

and

data
factor

the

..
Stimulation

falloff

stimulation

]

bulk-rormation

from

damaged

due

skin

visualized

wf )

is neither

equal

determined

not

the

a

of

its permeability

quantitatively.

-.damage
h (p -P

For

-~

fraction

shows

or

curve
141

kJ

section

analyzing

obtained

)

ft.

sampled;

Damage

buildup
stimulation

to

been

Well

from

data

significant

ha~

This

period,

:

a

area

curve-

which

is reached

short

confidence,

analysis

=872

a~

are essential.

from

Eq.

pseudosteady-state

viewed

Thus,

(i.e.,

probably

kJ'

From

the
hclp

estimates
analysis

and

helpful

average

if

with

permeability
type-curve

matching)

be

at
may

x 10

,

2.8

be determined

bulk-formation

can

MTR

should

clear-cut

cannot

quantitative
It

estimates

of

its plausibility
only.

If there

that

end

the

Y2

r;=302(6)

and

of

hours

50

Radius-of-investigation

beginning

] Y2

end

rt

,

k=162.6-;;;h.

-s

(7.65)(6)

qBII.

I

and

= 6 hours,

-302
-,
and

start

is

achieved

<l>p.c,

at dt

calculatc
estimate

from

I

at the

(948)(0.039)(0.8)(1.7

;.\' an apparcnt
the
middle-time

region

Y2

I

r; =

that

investigation

.

MTR
Thus,

MTR;

of

1.47,

kt

technique.

J r thcrc
slope
of

Eq.

r; = ()

earlythe curveand

the

radius

transient

From

MTR

nonlinear,

fitt~ng
using

during

by the

shut-in

MTR.

afterflow
effects
disappear.
the
probable
end
of the

mIddle-tIme
ve!i fied ?y a data
deviation
on
matching

the

is as follows.

the
when
that

of

procedure

permeability

by estimating
2, Assume

the

radius

reservoir.

(2.4)
to

,determine

the

portion

of

the

We

recall

that

PI

hr is the

~~

value

of

Pws

at shut-in

-

PRESSURE
BUILDUPTESTS

-31

"...;-~.;.;tI

~~. -I

Pws

I

0

:

in

,

I

C-

I(XX)

..:G-

p-.

J

I»"

EffiC1$

t = I hr

I
I/)

~

I

01

I

G

m

log

t1te, hr
Fig. 2.12-Log-log graph of examplebuildup test dala

t p +.1t
At

Fig. 2.13-Determination of PI h,.

time AI of I hour on the middle-time line, or its
extrapolation as shown in Fig. 2.13. It is not possible
to calculate the skin factor until the middle-time line
has been established because values of k, m, and
PI hr are found from this line. If an accurate skin
factor is to be calculated from a buildup test, the
flowing pressure PWf mu~t be me&1surcdbeforc ~llutin.
Interpretation of a given numerical value of the
skin factor can be summarized as follows.
I. A positive skin factor indicates a now
restriction (e.g., wellbore damage); the larger the
skin fal.:tor, the more severe the restriction.

pleting our consideration of skin factor solely caused
by formation damage or stimulation.
We now turn our attention to methods for translating values of s into less abstract characterizations
of the well bore. We consider three methods:
estimation of effective wellbore radius, 'h'U;
call.:ulatioll of additional prcssurc drop near the
wcllbore; and calculation of now efficiency.

2. A negative skin factor indil.:ates stimulation; the
larger the absolute value of the skin factor, the more
effective the stimulation.
3. Conditions other than well bore damage can
cause an apparent skin factor. The reason is that any

To ulldcrstand thc siglliricallcc of this quantity,
note that from Eq. 1.11,
q81J.
I 688 4>c ,2
pj-Pwf=
-70.6-fln('
IJ. ( w)-2S]
kh
kl

":SCimaCion
or ":ffccCive Wellbore Radiu~
The effective wellbore radius 'I~'Uis defined as
--s
'wu -, we
(2.14)

deviation from purely radial now near a well, which
results in to!al wel~ production squeezing through a
smaller vertical thickness near the well than away

--70
-.kh

6~

11n

( 1,6884>IJ.C('~
kl
)

from the well, increases the pressure drop near the
..
wl.'ll. This is precisely tIle ~alIICI.'ffcct that wcllborc
+In (e-2s) I
damage has; damage also results in an increased
pressure drop near the well. The basic equation used
-2s
in constructing our theory of pressure buildup and
= -70 6~
fln( 1,688 4>IJ.C
(~e
falloff test behavior, Eq. 1.7, is based on the
.kh
kt
)1
assumption that flow is radial throughout the
drainage area of the well up to the salldfac~; a
q81J.
I 6884>IJ.C,2,
deviation from this assumption invalidates the
= -70.6-ln'
(~ ) .
equation, but Eqs. 1.11 alld 1.16 arc u~U,llly CKl.:l.:llcll1
kh
kl
approximations when th(: nollra<.Jiall"low OI.:I.:Uf!i
Ilcar
th~ well bore only.
l11is shows. that the efrel.:t of ~. on total p~cssurc
Conditions leading to nonradial flow near the
drawdown IS the .same as that o~ a well wIth no

(

wellbore include (I) when the well docs not completely penetrate the productive interval, and (2)
when the well is perforated only in a portion of the
interval (e.g., the top 10 ft of a 50-ft sand). In these
cases, the analyst will calculate a positive skin factor
even for an undamaged well. (In addition, the
perforations themselves-their size, spacing, and
depth -also can affect the skin factor.) We will
examine results of this non radial flow after com-

alt~!cdlon.ebut wlth.aw~llboreradlusof"ru'.
.
C~ll.:ulatlon of effel.:tlv~ wellbore r~dlus IS. of
special value for a.nalyzlng wells wIth ve~tlcal
fracture~. Mod~1 studIes have s~own that for hIghly
c~nductlve vertIcal fractures with two equal-length
wlllgS of length Lf'
Lf=2'wo.
Thus, calculation

(2.15)

-

of skin factor from a pressure

buildup or falloff test can lead to an estimate of
fracture length -useful in a post fracture analysis.
However, this analysis technique for a fractured well
is frequently oversimplified; more complete methods
are discussed later.

producing about twice as much fluid with a given
pressure drawdown as it would had the well not been
slimulated.
Use of the skin factor method is illustrated in
Example 2.4.

Calculation of Additional Pressure Drop
Near Well bore
We defined additional pressure drop (t:\p).~ across the
altered zone in Eq. 1.9 in terms of the skin faclor s:
n
I
(AI})s=141.2~-~s.
::
kh
r In terms of the slope II' of lhe middle-time line,
&
( "'YJ
An
) s = 0869
( s,) ..."""".,...'
(216)
r
.,"
,

Example 2.4 -Damage
Analysis
Problem. Consider the buildup test described in
Example~ 2.2 and 2.3. Make the following
calc\llation~ with lho~e data.
I. c.'alculatcthc~kinfactorforthcte~tedwell.
2. Calculate the effect ive wellbore radius r M'Q'
3. Calculate the addilional pressure drop near the
wellborecausedbythedamagethatispresent.
4, Calculale the now efficiency,

Calculation of lhis additional pressure drop across
lhe altered zone can be a meaningful way of translating the abstract skin factor into a concrete
h
"
f h
d
II F
I
c aractenzatlon 0 t e teste we.
or examp e, a
100 STB/D
1 'tl
II
b
d .

5, Verify the end of wellbore storage distortion
using Eq. 2,13.
S I
0 II Ion.
t
I th
k'
f
'
1,InSk
rOC or.
n
e s In actor equatIon,

we
d

may
d

e
f

pro
1 000

raw
...Ine own 0

mIght

show

"

pSI

of

the
Th

zone.

the

I.
Imp

could

01

pSI,

A

I

'

f

na YSIS0 a

b

WI

1
t

. Id

a
t

UI up es

(t:\p)fls900psland,thus,that900

total
'.
IS

well

,

that

.

uclng
.

drawdown
h

les

t

produce

'f
at

I

h
t

occurs
d
e

much

the

altered
d

amage

were

nuld ..=

more

remove,

wIth

the

Calculation of Flow Efficiency
f

The final method that we will examine for translating
swell
intois abyphysically
characterization
calculationmeaningful
of the now
efficiency, E. ofWea
define now efficiency as the ratio of actual or observed PI of a tested well to its ideal PI (i.e., the PI it
would have if the permeability were unaltered all the
way to the sand face of the well). In mathematical
terms,
J
E= ~
Jidcnl
j

"

,

,.,(2.17)

need

..,.,
,the

hr

from

u

Ig..

0

'

.

extrapolatIon

of

the

mIddle-time

pSI,

.

u

=

e

ml

0

e

.

,

e-

,rom

an

Ime

ow

I

Ine

ex

0

eren

rapo

IS

IS

IS

Ime

UI
A.

=

I

.

I t
a

Ion

PI

hr

, f rom

th

e

actual pressure at 6t = 1 ~our: 4,103 pSI.) Then,
becausekl<t>IJ.C,
= 1.442 x 10 ,
(p t hr -P wf)
k
s = 1.151
-log"
) + 3.23 J

l

(

m

<t>p.c
I'W

f (4 295 -3 , 534)
= 1.151{'
70

-IOg/

rapid analysis of a pressure buildup or falloff

p

PI

t 0 a s h u t -In'
tIme
'
0 f
I
h our.
At
t
ho
(t + A t)/ A t 13631 F
(F' 2 I PI ) f th
ddl t
I'
t th ' t '
4 295 ' (N t h
d ' ff
t th ' '

1.442 x 107
98 2 1+ 3.23J = 6.37.
(0. I )

rM.u=rwe-'~

,

of

the

mIddle-tIme

,-,

lIne

to

tllclli~p-P\l'rinEq,2,17con~tant),
flow efficiency is unity for a well that is neilher
dal!l~ged I~or stimulated, For a d,amaged well, now
erflclency ISIcss than one; for a stlmulatcd well, now
efriciency is grcatcr, t!lan one, A d~magcd ,,:ell wilh a
calculated now efflclcncy or 0.1 IS producIng about
10% a~ much nuid with a given pressure drawdown
a~ it would if the damage were removed; a stimulated
wcll with a calc\llated nO\V efficiency of two is

---~

..

= (0.198)e -6.37

..

0.00034 ft.
The physical interpretat ion of this result is that the
tested well is producing 250 STB/D oil with the same
pressure

extrapolallon

(t,,+6t)/6t=I,lsfoundmorereadllythanp,whlch
, I
I
I . FI
ff '"
can requIre cngt lY ana YSI~, oW e Iclency IS actually time dependent unless a well reaches
p~cudo~teady ~tate during the producing period (only

,
~

"

Iwe
.

P-Pwf

test, Eq. 2.17 can be written in approximate form as
.=
P -P"f(l1p)s
E=.
' ..".,...,..".,
(2,18)
P -PM:!
wllere

.I:'

In Example 2.2, we found from curve matching that
s:5, which is good agreement.
2. I:.jfedive U'elllJore Radius. From Eq. 2,14,

=P-P~if-(/1f)s.

.For

..

ur,

across

same drawdown or, alternatively, could produce the
same 100 STB/D with a much smaller drawdown.
'"

.

d'

drawdown

f 0 00034 ft

ra IUSO,
th esan df ace,

3, Additiol,al
I:rom Eq, 2,16,
(~)

-0869
S-,

as

an

would

d

a

permea

well

b 'l'

with

a wellbore

II t y una It ered Up to

Pre.5sure Drop Near tl,e U-"elllJore,
III ()S

=(0,869)(70)(6,37)
= 387 si.
p
Thu~, of the total drawdown of approximately
4,420 -3,534 = 886 psi, about 387 psi is caused by
damage, Much of this additional drawdown could be

-

PRESSUREBUilDUP TESTS

~

"

~:~

:-

33

avoided if the skin resulted from formation damage

I

-

(rather than from parlial penetration, for example)
and if the well were srimulated.

Example 2.5 -Incompletely
Perforated
Interval'

4. F/ow Efficiency. To calculate flow efficiency,
we need p., the value ofpws on Ihe middle-time lille
at (tp+At)/At=
I. We cannot extrapolate dircl.:tly
on our plot because there are no values of
(t p + AI) I At less than 100, bur we nOle that the
pressure increases by 70 psi over each cycle; thus, we
can add 2 (70) psi to the value of P at
(lp+At)/At=IOO:

Problem. A well with disappointing prodllctivily i~
pt:rforated in 10 ft of a total formatioll thil.:kllC:~Sof
50 fl. Vertil.:al and horizolltal permeabilitic~ "art:
believed to be equal. A pressure buildup lesr wa~ run
Oil the well; results and basic propcrties are ~ummarized as follows.
I 190 .
PIYj='
ps~,
PI hr = 1,940 pSI,
<p = 0.20,
m = 50 psi/cycle,
II. = 0.5 cp,
rw = 0.25 ft,
CI = 15 x 10-6/psi -I, and

p' = 4,437 + 2 (70) = 4,577 psi.
Then, from Eq. 2.18,
£=P.-Pwj-(.:1p)s
p' -Pwj

k = 3.35 md.
=

4,577 -3,534 -387
4 577 -3 534
,
,

Calculate s, sd' and s p; on the basis of these resulls,
determine w~ether the productiviry problem results
from formarlon damage or from orher causes.

= 0.629.
Solution. From Eq. 2.4,
-k
S=I.151[Plhr-PWj
-Iog(

This means that the well is producing about 621170
as
much fluid with the given drawdown as an undamaged

well

in a completely

perforated

would produce.
5. £ndof Wet/bore Storage Distortion.
2.13and Examples 2.2 and 2.3,
I

interval

From Eq.

Cs eO.14s
= 170000
'

wbs

kh

=

<PII.Clfw

]

= 1.1511 (1.940- 1,190)
50
-

-10

III.

(170,000)(0.01 1S)e(O.14)(6.37)

2)+3.23

m

(

3.35

g (0.2)(0.5)(1.5xI0-S)(0.25)2

) + 3. 23 ]

= 12.3.

(7.65)(69)/0.8

.The

=7.42 hours.
This agrees closely with Ihe results of Example 2.2.

~

I.:ontribution

<:>fan in~omplelely perforated

Interval to the total skin factor IS, from Eq. 2.20,
h
h m
~/J= (f
-1)[ln(/
V~
)-2]
p

w

V

Effect of Incompletely Perforated Interval
When rhe complered interval is less rhan total for~ation rhickness, the pressure d~op near Ihe well is
Increased and

the

apparent

ski!)

factor

becomes

increasingly positive. In a review of technology in
this area, Saidikowski6 found thai total skin factor,
5, determi.ned from ~ pressure transient re.stis related
10Irue skin factor, sd' caused by formation damage

and apparent

skin

factor,

sp'

completely perforated interval.
betweenthese skin factors is
hi

S=j;-Sd+Sp'

caused

The

by an in-

relationship

(2.19)

p

where hi is total interval height (ft) and hlJ is the
pcrforaled interval (1'1).
r

Saidikowski .also verified that Sp can be eslimated
rrom the eqllatlon
h
I ( h
r-kI
~"J= (-L -I) In -Lv
---}l- ) -2,
(2.20)
hp
rw
kv
where k if is horizontal permeability (md) and k v is
vertical permeability (md). Use of these equations is
b~stillustrated with an example.

(

50
= 10 -I

)[In (~v50

~~

) -2 ]

.l

= 13.2.
Rearranging Eq. 2.19, skin factor, sd' resulting from
formation damage is
h

:

l
!
f

Sd= :.:2. (s-sp)
hi
10

=-(12.3-13.2)
50

= -0 IS
..
A~ a practical matter, the well is neither damaged nor
stll1\lllaled. The obscrvcd produclivilY problem is
,-"all~cdcnl!rcly by Ihe effc,-"l~ of an in,-"omplclcly
pcrloruled Inlcrval.

~I
;,i
~
f

Anulysis of Ilydruulicully "'ruclured Wells
Type curves provide a general method of analyzing
hydralllically fractured wells -particularly
because
~

Sla'E

E.~
PWS

FR/(;~E

T

TABLE 2.4 -BUILDUP

DC».4INATB
ClRVESHAPE

HYDRAULICALLY

TEST SLOPES FOR

FRACTURED

L,lr.

mm..'m'ru.

0.1

0.87

0.2

t

+6t

WELLS

0.70

0.4

0.46

0.6
1.0

0.32
0.28

logT
Fig. 2.14 -Buildup
curve for hydraulically
bounded reservoir.

fractured

well,

finite condl,lctivity can be considered. Some convention~1 .methods are a~so o~ value for infiniteconductIvIty fractures. ThIs sectIon summarizes some
of the useful conventional methods.
When fractures are highly conductive (i.e., when
there is little pressure drop in the fracture itself) and
~hen there is ulliform nux of nuid into the fracture,
lInear now theory de~cribes well behavior at earliest
t~mcsnow
ill a rates
buildtlp
te~t. (Uniform
nux
mcall~
idclltical
of
fonnation
nuid
illtO
thc
fracturc
..w

pcr Ullit cro~s-sectlonal area at all points alollg the
fracture.)
From Eq.
1.46, for collstallt-rate
product ion,
I
qB

IJ.f

Thus,
L
L
-210g( ~ ) = -2 log ~ -210g
2r w
2
= (PI hr -Pwj)
m
-210g (

(~

-IOg

(~

r w)

)

<l>IJ.C,

~ ) + 3 23

r.

.

This simplifies to
log ( L j) = ~

V2

2

.

l(~

mI

~)

) I
11Lj k<l>c,
k
f Forabuilduptest,forlp)..ill,
+log;;;;;--2.63.
(2.22)
qB
ill V2
.'
Pws -Pwj=4.064
-(£.) .~slng
Eq. 2.22, fracture I~ngth, LI' can be estimated
hLj k<l>Ct
if the MTR can be recognIzed, whIch allows m,PI hr'
Thus, the slope ',1 of a P vs. v'A7 plot is
and k to,be determined.
L
wYi
In buIldup tests from some hydraulically fractured
) 2.
(2.21)
:-veils, the ~rue.middle-time line does not appear, as
"'L =4.064 ~
f
IlL f k4>c,
Illustrated In FIg: 2.~4. (~fter~ow can cause the same
From measurements of this slope, fracture length,
c.urve shape.) ThIS sltua.tlon a.nse~bec~u~e, at earl~est
/4f' can be estimated. This procedure requires that an
tlme~, the depth of InvestIgation .'s In a region
independent estimate of permeability be available -d°m.lnate~
by. the fracture; at lat.er tImes,. the.depth
from a prefracture pres~ure buildup test on the well,
of Investigation reaches a point dominated by
for cxamr;le.
boulldar~ effect~. (Se~ Fig. 2.14.) When the length L,r
When linear now cannot be recognized (i.e., when
of ~ vertical rracture IS greater than one-tenth of the
there is f10 _early straight-line relationship between
draln~ge radius r e of a7well centered in its drainage
p",Sand v'A1), we can use the observation that L
area, It ha~ been found that boundary effects begin
= 2 r, for infinitely conductive fractures to estimat{
before the Innuence of the fracture disappears. For a
fractl~'~elellgth. Rather thall calculate s directly, we
given drainage radius, the greater the fracture length,
can note that
the greater the discrepancy between the maximum
slope achieved on a buildup test and the slope of the
s= 1.151 1 (PI hr -Pwj) -IOg ( -~
) + 3 23] .true
middle-time line. Table 2.4 summarizes the ratio
nl
<l>JA.c,r~.'
of the maximum slope attained in a buildup test to
and, bccau~c
the ~Iope of the true middle-time line (from the work
L
of RII,~~cll and Truitt')
for infinitely conductive
,
= ~ =r e-s
fracturcs.
"'0
2
w
,
The implication is that if the test analyst simply

~

-~

Pi-PI~1=4.064-(-

1

(~

then
s= -In(

does the best he can, and finds the maximum slope
on a buildup test from a hydraulically fractured well
and ass.ume~that this maximum slope is an,adequate
approxImation to the slope of the true middle-time

L
L
~ ) = -2.303 log( ~ ).
2, w
2r w

':.

~-

PRESSURE BUILDUP TESTS

35

ETR

ETR

Pws

MTR

""

t-p.

MTR

P
I

L TR1/

RI

I

ws

~...

log

tp + .1t

--.11---

Fig. 2.15-Buildup test graph for infinite-acting reservoir.

log

Fig. 2.16-Buildup

tp + .1t
--~t

-/

test graph for well near reservoir

limit(s).

line, then the permeability, skin factor, and fracture
estimates will be in error, with the error growing as
fracture length increases.
Correlation of reservoir model results by Russell
and Truitt 7 showed that an equation similar to Eq.

For a reservoir with one or more boundaries
relatively near a tested well (and encountered by the
radius of investigation during the production
period), the late-time line must be extrapolated (Fig.
2.16). (This can be quite complex for multiple

2.22 can be used to estimate true fracture length even
when L f > 0.1 r e.
We again emphasize that all methods in this
section assume highly conductive, vertical fractures
with two equal-length wings. When fracture conductivity is not high, fracture length estimated by
thesemethods will be too small.
.For
2.9 Press~re Level In.
Surrounding
Formation
A pressure buildup test can be used to determine
average drainage-area pressure in the formation
surrounding a tested well. We have seen that ideal
pressure buildup theory suggests a method for
estimating original reservoir pressure in an infiniteacting reservoir-that
is, extrapolating the buildup
t~st to infinite shut-in time (tp+dt)/dt=IJ
and
reading the pressure there. For wells with partial
pressure depletion, extrapolation of a buildup test to
infinite shut-in time provides an estimate of p.,
which is related to, but is not equal to, current
average drainage-area pressure. In this section, we
will examine methods for estimating original and
current average drainage-area pressures.

boundaries near a well.) Note that our discussion is
still restricted to reservoirs in which there has been
negligible pressure depletion. Thus, even in the case
under .consideration, the well must be relatively far
from boundaries in at least one direction.

Original Reservoir Pressure
For a well with an uncomplicated drainage area,
original reservoir pressure, Pi' is found as suggested
by ideal theory. We simply identify the middle-time
line, extrapolate it to infinite shut-in time, and read
the pressure, which is original reservoir pressure (Fig.
2.15). This technique is possible only for a well in a
new reservoir (i.e., one in which there has been
negligible pressure depletion). Strictly speaking, this
is true only for tests in which the radius of investigation does not encounter any reservoir
boundary during production.

Static Drainage-Area Pressure
a well in a reservoir in which there has been some
pressure depletion, we do not obtain an estimate of
original reservoir pressure from extrapolation of a
buildup curve. Our usual objective is to estimate the
average pressure in the drainage area of the well; we
will call this pressure static drainage-area pressure.
We will examine two useful methods for making
tllese estimates: (I) the Matthews-Brons-Hazebroek
(Mllll)8 p. method and (2) the modified Muskat
l1Il:thod.9
The p. method was developed by Matthews et al.
by computing buildup curves for wells at various
positions in drainage areas of various shapes and
then, from the plotted buildup curves, comparing the
pressure (/).) on an extrapolated middle-time line
with the static drainage-area pressure (p), "r'hich is
the value at which the pressure will stabilize given
sufficient shut-in time. The buildup curves were l:omputed using imaging techniques and the principle of
superposition. The results of the investigation are
summarized in a series of plots of kh (p' -p) /70.6
q/lB vs. 0.000264 ktplt/>/lc,A. [Note that kh(p. -p)
/70.6 q/lB can be written more compactly as 2.303
(p* -p) 1m. Also, the group 0.000264 ktp/t/>IJ.C,Ais
a dimensionless time and is symbolized by t DA .The
group kh(p*-p)/70.6
q/lB is a dimensionless
pressure and is given the symbol PDMBHJ. The only
new symbol in these expressions is the drainage area,
A, of the tested well expressed in square feet. Figs.
2.17A through 2.17G (reproduced from the Mat-

Chapter 6

Other Well Tests

6.1lntroduction
This concluding chapter surveys four well-testing
techniques not yet discussed in the text: interference
tests; pulse rests; drillstem lesls; and ~'ireline formation rests. These tesls and olhers covered in
previous chaplers by no means exhausl the subjecl;
however. the comprehensive Irealment needed by the
practitioner is provided by SPE monographs 1.2 and

In an infinite-acting, homogeneous, isotropic
reservoir. the simple £i-function solution to the
diffusivity equalion describes Ihe pressure change al
Ihe observation well as a function of lime:
qBIL
IP/LCr
Pi -Pr = -70.6kh£i(
-948 -t).
...(6.1)
1

Ihe Canadian gas well tesling manual.3
.The
6.2 Interference
Testing
Interference tests have two major objectives. They
are used (I) to determine whether two or more wells
are in pressure communication (i.e., in the same
reservoir) and (2) when communication exists, to
provide estimates of permeability k and porosity/compressibility product, d>£i, in the vicinity of
the tested wells.
An inlerference test is conducted by producing
from or injecling inlo al least one well (the active
well) and by observing Ihe pressure response in at
least one olher ~'ell (Ihe observalion well). Fig. 6.1
indicates the typical lest program with one active ~'ell

This is simply a restatement of a familiar result.
pressure drawdown at radius r (i.e., the observation ~.ell) resulting from production from the
active well al rare q, slart1ng from a reservoir initially
at uniform pressure Pi, is given by the £i-function
solution. Eq. 6.1 assumes thaI the skin factor of the
aclive well does nor affecl the drawdown al the observation ~'ell. Wellbore storage effects also are
assumednegligible al bOlh Ihe aclive and observation
wells "hen Eq. 6.1 is used 10 model an interference
test. JargonS shows thaI bOlh Ihese assumptions can
lead 10error in testanalysis in some cases.
A convenienl analysis lechnique for interference
lesls is Ihe use of Iype curves. Fig. 6.3 is a type curve
presentedby Earlougher; I it is simply the £i function

and one observalion well.
As the figure indicales. an active ~'ell starts
produl.:ing from a reservoir at uniform pressure at

expressedas a function of its usual argument in now
problems. 948 oJJ.'ir2/kl. Note that Eq. 6.1 can be
expressed complelely in terms of dimensionless ..

Time O. Pressure in an observation well, a dislance r
a\\ay, begins to respond after some lime lag (related
to the lime for the radius of investigation
corresponding 10 the rate change at Ihe active well to
reach the observation well). The pressure in the active
well begins to decline immediately, of course. The
magnitude and timing of the deviation in pressure
response at I~e observ~tio~ well d~~?ds on reserv?ir
rol.:k and fluId propertIes In the VICInity of the actIve
and observation wells.
Vela and MI.:Kinley~ showed thaI Ihese properties
are values from the area investigated in the test -a
reclangle \\'ith sides of length 2ri and 2ri + r (seeFig.
6.2). In Fig. 6.2. ri is the radius of investigation
achieved by the active well during the t.esland r is the
distance bel ween active and observallon wells. The
essenlial point is thaI the region investigated is much
grealer than some small area bel ween wells, as intuition might suggest.

variables:
I
Pi -Pr
= --£i
qBIL
2
(141..!~)
.!

I
1(-

-)
4

2
".
QILC,r'"
)( ~ ).1,
0.000264 kl 'r..,
,

or
I
PD = --£i
2
\vhere

2
( ~),
41D

-kh
PD = (p, Pr),
141.2qBIL
rD =rlr ""

, .-_(6.2)

PRESSURE

---=~lr";'

BUILDUP TESTS

~~fltff~-

_~i;""

, r -~ --1- -~ -l!
,

1---

;: 1i,

I

37

I

!

6

.
I ~~

:

:

I

.
.

I

!

I

-.

I

c.'m
II
.,a

i

~

I;

!

I

I

':

'

I

i
I

:
:

I

I
i

i
:

I

:
i

i

i

/

m
/1
u..o-;;;;

!

I,
"

..'1"

/1..

//

/1'

.'

I

I

""

/"

-'

:

.'

0

""

,
tIJ,

1 " ",-

--'

c.jCD:

-1

/1

::

--1
1

1

:

/1

i

~ I "'"

0,01--r-

01 I 01: 04

06 ~ 01

1-"'I

I

.
..1

I

I'

6 ..

0,000264 kt

~,.cA
Fig,

2.178 -MBH

pressure

function

for

differenl

0.<XX>264kl pss
IDA =

t;I£C,A

well

locations

in a square boundary,

general drainage-area shal'N.'Srollow eq,.ations of Ihe
form

=0.1,

or. for this case (with A = 160 acres =6.97 x 106 sq

P-Pwj=

141.2~

ft)..

kh

1~In(~~
~)2

CAr.

-

~ +s.
J

4

Ipss=183 hours.
"

(1.20)

The reader can verify that use or I ss in the Horner
plot and in the p' method leads to t~e same results as
in Examples 2.3 and 2.6.
.vs.
Shape Faclors ror Re~ervo.rs
Brons and Millert2 observed that reservoirs with

;;

after pseudosteady state now is achieved. Values for
the shape factor CA can be derived from thePDM8H
I D.~ charts for the various reservoir geometries in
Figs. 2.17 A through 2.17G. Note (hat (he definition
of p' implies that

i
f
I

I

!,

II'.-.I

f

I

ii

p -p

:-'

~

,!

706QjJ.B/kh

.I:

I

"
:~

---!,

:

5

--~

-':- ,.,1/ ""

4.

;\

/

;

~~..-1

~ ,,~

3
"~~--:--

.-1""

2.

-"""",:

/,

"
I

""

""

~

~

:'

~ ~

/

---,

[

/'

'

~,..,.

i
I

0
01

--""
02

03 04

:'
061

QI

2

3

0.000264

4

,

I

2

3

4

,

10

.I

kt

+jJ.cA
Fig,

2.17C -MBH

pressure

function

for

different

-~__~~JI_II..

well

locations

in a 2:1 reclangular

boundary

38

-WELL

~

;~~-r-]

TESTING

::rrrj--"-

706Q1/.B/kh;:i!':i

4

1---;
!

'

;-!"1;--'---'

!

I

,~I

!

.2

I

1

0

I

-I

-2
06

01

2

3

4

0000264

6

1

2

3

.6

10

kt

~1/.CA
Fig,

2.17D -MBH

f

p' -P":f

q 8 IL

=70.6T

pressure

function

for

different

well

locations

In(tp +~t)/~t,

~

h
and lhal, allhe instant of shut-in (~t = 0),

P.q81L
-p"j=70.6-

kl

f In
I

-In

,~,688tPlI.£."",

I

(

k/1

kt

CA'",

-q8Jl

0.<XXJ264ktpCA

-70.6kj;-

acting reservoirs only.
Eliminating P"f between equations,
p' -jJ = 70.6- q8p. In

( 10.06
A
,... _2 ) + 1.5]

) +2s. I

( -~-:-::-:!
kt

These relationships result from replacing Pi with p'
in Eqs. 1.11 and 2.1, which are valid for infinitei

boundary,r

in a 4:1 rectangular

In(

tPlJ.C
A
I

)

I
:

- 70 6!!!!!:I

)

-.n

i~6si:;~~.

kh

(C

)
At

DA

l
'

.

0r
I,

,

..!

i
i

.--I
-!r

!

~I

:

--'
loCo

.,

,oX

a,ljQ'
II ~
.' C'

'

I

.

0.1 co
I

o.

,

.,

,...

,

,

"

:.

...,
Fig. 2.17E -MBH

-

M
pressure

I'
...,
0.000264kt
function

~llcA

for

rectangles

.G
of

various shapes.

r

.c
~

PRESSURE
BUILDUPTESTS

I

roo

d

~

Q.m
.a

0..0

4

3

2

-39

:

;

10

4IllcA

:
,

O.OCXJ264
kip
cPlJ."
,A

.o'

10

= I,

GO

100

,.

,'

.'

!fit

~

p. -p
PDMBIJ = 706 8 /kl =3.454.
.q IJ. I
Thus, In (C A (I») = 3.454 or C A = 31.6 -essentially

IDA =

determine CA. For example, consider a circular
drainage ~rea with a centered well.
From Fig. 2.17A, for

Fig. 2.17F-MBH pressure function in a squareand in 2:1 rectangles.

p. -p
PD MBH = 70.6 q81J./kh
=In(CAIDA)'
This equation implies a linear relationship between

f.

~

.c
I

01

0OOO264kl
.,.cA

P D MBH and IDA after pseudosteady state now has
been achieved. Indeed, inspection of the curves in
Figs. 2.17A through 2.17G shows that, for sufficiently large IDA' a linear relation does develop.
Further, any point on this straight line can be used to

,

;1';
ICI.~

-I

-3

001

0 0
po.

\D

.1 ;;
Q.

:,; ;
I

1Oi

"

i:;;!

.j...
!;.,

III
;'
t
.'!!o.-""

!' :\~; 0

.0'

-':

tu',r;
.-2

"

"):1'
r

~ ~'

;~'

Fig, 2.17G-MBH pressurefunction on a 2:1rectangleand equilateraltriangle.

,.;:;~:;:,~:;,.

'

,.

0

:
'

I

iI
!',

I
[

..

ii.

"C;L.L

TABLE

2.5 -APPLICA

30

4.393

40
50

4.398
4.402

72

4,407

TION OF MODIFIED

--15

(ho.;}~rS)~;i)
60

MUSKA T METHOD

4.405

~

19

29

-;,

10
6

14
10

24
20

~

17

I

1

5

15

'e::.

-i;4.408 o;;~~1f;~~=4.422
3

7

thc val'Ic

given

rclatioll...hip

in Table
between

11>..t ~O.I;

ill Table

1.2, this

p~cudosteady-state
the column

J .2. Note
P/)~tmf

now

"Exact

al~o
and

that

for '0/1

linear

begills

is the value

equation

the

IDA

cylindrical

(in

> ").

At

position
now

to

of

boundaries)
are

felt

at

simulate

(depth
in

proximated

Fig.

which
is a solution
to the now
well
producing
from
a closed,

re~ervoir

constant

rate.

a buildup

and

noting
buildup,

Using

rollowing

investigation

the

has

that,

once

the

boundary
can

establish

effects

data

be

provide

ap-

-0.00388

k4//</>IJ.c,~).
~
(2.23)

F
..drainage
.or a~lalysls
of buildup

I

tests,

we

usually

express

t~is

as

)

QBIJ.

118.6kh

0.OO168k111
-</>p.cr2.
, ~

ill tllc

used
..II'lt-ill

2~f) ""(',r;

tllat

this

equation

log<l;

-PI\:f
A

cq'Iatiol1
value
straight
p (thc

p

line

form

tested

be

that
i~ too

ha~ a noticeable

in the time

a~sumed

long,

We

illustrate

now

2.lft.)

in (and

the

only

important
requires

has

to

portion

of

found

to

been
for

with

hydraulically

layers

only

of

different

at the wellbore.

fails.
The
too:
(I)

reasonably

Muskat
it fails

centered

drainage
area
in
derivation

required

shut-in

times

are

frequently

this

in

method

in its

need
of

</>Jl.C,r;)/k
particularly

no

it is used

of

not
this
(250
im-

low-permeability

with

an example.

2.

assume

a

Solution.

a

method

value

of

well)
in-

outside

of

we do

value

have

vs.

for data

not

To

well's

Ca/cli/a/e

in Drainage

Consider
2.2 and

the
2.3.

A rea

buildup

test

Estimate

drainage

area

described

the average
by

using

in

presStire

the

modified

method.
The

data

are those

the

of

time

not have

limit

that

the

can

in the range

</>IJ.C
,r;)/k.

estimates

slightly

Modified

Pressure

111 = (750

tested

7-

In this
k and

range
these

r~,
of

complicales

(250

by

to

fortunate

to

Often,

-a

trial-and-error

data

of course,

situation

method,

this

</>IJ.c,r;)/k

so we can eliminate

interest.
of the

the

examined

case we are

estimates

applicability

be
6./=

that

does

but one
nature

lhat

of the

calculations.-Here.
750 4tIJ.C,r~
k'

~11/~

value

the

method

curvature

(750

Me/hod

Muskat

111 until

-PII'f)

(2) the

practically
reservoirs.

the

assumed

well

to

of the

of log(jj

downward

of

p that

noticeable
upward
curvature
l11e correct
assumed
value
litlC

An

a plot

of

range

2S0 4>#(clr~
--~

An

the correct

with

sel1sitive.

producc!i

We
vs.

pressure

Experience

it is quite
low

applied,

it does,

drainage-area

p that

is

10g(p-PM's)

when

found.

dicatc!i

I

it

plot

form

wells

(although
the
as implied

and

in
This

it

communicate

</>IJ.c,r;)/k

Problem.

constants.

how

and
results;

!itatic
been

6.( that

are

suggests
for

has

8

for

it

correct

jJ estimates

and

area
cylindrical,

Mliska/

) = /1 + 8111,

and

the

is not

the

Examples
whcrc

choose
(2)

two

when

when

A verage

2.24 has the

(I)

p.
method
disadvantages,

Exalt'p/£,

~ 111~ -k-'
Eq.

has

method.

are

ral1gc

7~f) 4>/(.,r;

kNote

in developing

timc

method

and

that

Muskat

method:

In these cases,
the
method
has serious

(2.24)
Approximations

to

analysis);
wells

lor modilied

properties

satisfactory

-method);

(

og(j>-PII:f)=log

valid

for

p.

reservoir

permeability

= 118.6~exp(
kh

the

jJ (except

fractured
B

equation

of

graph

Muskat

over

estimates

reservoir

equation

modified

advantages

stabilized

reached

2.18 -Schematic

The

super-

as

P-PII'.f

LON

the

exact

.

Eq.
1.6,
for
a

P

CORRECT
15

TOO

Modiried
Muskat
Method
k
h d
b
d
I ..,
The
d ' fi d M
mo
I Ie
us at met 0
IS ase
on a Imltlng
form
of
equations

ASSUMED

-ASSUMED

at

at which

becomes

IINI,j

ASSUMED
TOO
HIGH
p

~
~

g

II;;;)

in) the

250 4tIJ.C,r~
k

is too

high

in this same
of p produces

proper

time

range.

produces
time range.
a straight

a

---

=

(250)(1,320)2
(1.442x
107)

=30.2

hours,

and
750 </>C ~
IJ. , ~ = 90.6

(See Fig.

hours.

k
Thu~,

we can

~

examine

~-

data

from

4/=30

hours

B

until

BUILDUP

TESTS

41

I

!
TABLE

2.6 -BUILDUP

~(

DATA FOR WELL NEAR

Pws

(hours)
0

~(

Pws

BOUNDARY

~(

tOO

Pws

(psia)
3,103

(hours)
8

(psia)
4,085

1
2

3 , 488
3,673

10
12

4, 1 7 2
4,240

36
42

4 , 700
4,770

3

3,780

14

4,298

48

4,827

4

3,861

16

4,353

54

4,882

5
6

3,936
3,996

20
24

4,435
4,520

60
66

4,931
4,975

we stop
We

recording
make

From
clearly

pressures

Table

plots
the

2.5

at ~

for

of (p-Pws)'
best choice;

(hours)
30

(psi a)
4,614

This

example

using

the m~thod.

5 psi

of

in such

a case,

this

method

The

trial

are

of

value

significant

the

values

of

MBH

only

from

this

section,

we

the

Muskat

Fig.

2.19 -Modified

of

built

up to withil1
of

the

to

art.

The

problems

Both

by Earlougher.

than

is

well

to a single

In Chap.

I,

superposition
pressure
in

l

boundary

1.50.

'

when

(such

are

P;-Pw/'=

state

based

a single

boundary

near

a well

application
that
from

faull)

of

is given

by

can

develop

such

a well.

an equation
Note

cf>p.CI(2L)_
kip

[ -948

describing

]

k(1

--PRESSURE

can be written

Two

test

tuall~

for

k~

for

25)
the

the

Ei

becomes
1 +.-11
.p .--)

I +.-11
.p '--.)]

+ In(

~

.-11

( I'P' +.-11
-.. )

qBp.

sl
j

buildup
slope

Eq.

of

than

the

test.

For
a

for

with

For

large

approximation
time

Eq.

2.1)

.this

reason,

buildup

test

time

to

ordinarily

near

be valid

and

(2)

can

be
or

of

L

or

required
can

available

for

~waiting

a doubl~ng

IS

necessanly

not

a

will
~q. even.2.26

values

shut-in

(2.26)

a well

curve
fault,

slope to double
.JI.C,L 2 /kA/<O.02,

permeability,

logarithmic

on

(I)

2.26

for
the
3,792

tjlp.c,L2/k.

values
the

.,

slope
such of as a abulldu,P
s~ling

(~ompare

.-1/>1.9xIO'

longer

) ~kh

accurate

can be made:

boundary,
that the
double

small

cf>p.CE ) -2

that

as

observations

single
shows

+~)

[ In ( ~~

large

qBp.
Pws=Pi-325.2-log(lp+.-1/)/.-1/].
kh

p

q

is

qBp.
ws = 70.6 -=-Iln(

the time
r~qulred
long -specifically,

l

sufficiently

kh.-11

that

-706~rln
.kh

-70.6 (-

the equation

Eq.

,

ti.J

time

= 141.2-ln

[ ~~88cf>Jl.CA ] -2s- J
P, .-= Pws

tjlp.C L 2
I --).

the

]

a buildup

792
k~

approximation

Pi -P

kip

-70.6~Ei

for

shut-in

functions,

2

We

a

This

kh

1

(2
For

the flowing
a no-flow

"

...': -,"

-3
~-

kh

to double,
and
distance
from
a

qBp. [In ( 1,688cf>p.Clr~) -2s

kh

Bp.
-Ei(

( -q)

kh

we showed
distance
L

tjI~

on

and
basedis

it becomes

-70.6-

-70.6

logarithmic

we illustrated

;I

example'

k (Ip + AI)

of

t3

as a sealing

Rearranged,

the

Eil~92

kh

boundary.

principle,
a well
a

to

a

for

has
Much
been technology
developed

causes
the slope
of a buildup
curve
then develop
a method
for estimating
tested

applied

either

techniques

presented

summarized

that

melhod

methods

pressure

with

rather

techniques
data
test analysis.
analysis

demonstrate

hr

'

tesl.

-70.6~

briefly

on
buildup-test
drawdown

We

Muskal

buildup

t

70

stabilization.

deal

these

~

72 hours;

in applying
final

~
~

estimating
reservoir
size and distance
to boundaries.
These comments
are introductory
only, and deal only
with
the simplest
cases. The intent
is to illustrate
an
approach

p. 4412

I
M)

2.10 ReservoirLimits Test
In

~:::::~~~~~

K>

for

time

method.

when

distance

had
value

p'

;~.:~~--o
P' 4422

p.

the mechanics

at a shut-in
is little

the

modified

only

pr~ssurc

value
there

or

of

to illustrate

its static

~:=:~-Q~~~~
o

we see that p=4,412
psi is
we also note the sensitivity
of

application

is intended

;

I
10.-

= 72 hours.

three

this
method.
There
is a noticeable
curvature
p = 4,408
psi and p = 4,422 psi (i'"ig. 2.19).

method

(/)
0.- ~

be
a
in
a

I.

,; ~;..r:1CIO"~'~e~~oc
!,.

,

, "",
""', '-

\

.' '.\';

..."'.."p".".,,.
...\"..-'.
..."--"--"-'

:.
,.

Or ~cen!!fy;~~

:1 no-now..
o '"
."
:- "-~. ~'l)U!~Car\ "cr-t.'("'O,,"v
SO"""
"",f:t.'
2 '~-""',:."."""
'."""'-"""'\'~.".". '."-~': :'..

I'j,;"':"'r~:,~;:-".:;.~":'..':"~,.',.,
"'
,:?,
I-\r
.:,-;-

-'.;

..;

,

" --,,- .:\\\:.a~\:.es'.~'~~"':.::-:s~a!1\.".,j"..
,"
,
."
~.

(

",

.'.,,'

.,!.,

:,'

']

..-

.":'j

*
,'l\~

-'

P ws

-O,434£i( -3,792op.c(I_:,'
\

kIp

A

~;)

~

,

""~

1""""""'-

/:

( ~~~~~\ktlt

-70 ,6~£'
kJ1'

,V-rR

V

LTR.

L-

"'--~

I

,~ ~
J' ...~k,..7)

Reasons for arranging the equation in this form are
as follows,

I
09

tp + ~t
-~t--

1. The ~erm 162.6 (qB~/kh){log[(tp+~t)/4IJ
-0:~34 £, ( -3, 79~ lPp.c,.L2 / ~t p) ] I determines the,
position
middle-time
lane.
the the
Ei
flilidion ofis the
a constant;
thus,
it Note
affectsthat
only

FIg. 2.20- boundary,
Buildup lest graph lor well near reservo'Ir

position of the MTR and has no effect on slope.
2. At earliest shut-in times in a buildup test, Ei
(-3,7921Pp.c,L2/k~t)
is negligible. Physically, this
mcan~ that the radius of investigation has not yet,
cllcountered the no-now boundary and, mathematically, that the late-time region in the buildup
tc~tha~ not yet begun.
Thcse observations suggest a method for analyzing

.
d~llled well. ~o confirm this fault and to estimate
distance from It, we run a pressure buildup test. Data
from the test are given in Table 2.6. Well and
reservoir data include the following.

thc buildup test (Fig. 2.20):
I. Plotp"~vs.log(tJ!+~t)/~t.
2.
Establi~hthemiddle-timeregion.
J. Extrapolatc
thc MTR into the L TR,
~. Tabulate the differences, Ap~'S' between the
bu~ldup c~rve and extrapolated MTR for several
p<.)lnt~(At)I':(=P":f-PMT).
5. Estimate L from the relationship implied by Eq.
2,2ft:
.11':,:\ ~

70.6 fill'l

-1:.'i( --::_~~792
I/I/lc',1,2)).llic
k~t

IP =
Jl =
=
rCO
( =
Aw: =
Po =
q =
B: =
J1 =

0.15,
0.6cp
t7x ft10-6 psi-I '
05
0:00545 sq ft,
54.8 Ibm/cu ft,
1221 STB/D
1:310 RB/STB, and
8 ft.

,
(2.28)
,. i~ Ihc only unknown in this equation, so it can be
~ol\'cd dircctly. Remember, though, that accuracy of
Illi~ cqualion requires that ~t.cft .when this condil iOiI is not satisficd, a compuf~r history match
u~ing Eq. 2.25 in its complete form is required to
detcrminc I~.
111i~ calculation implicd in Eq. 2.28 should be
madc for several values of ~t. I f the apparent value
of I. tcnds to increa~c or to decrease systematically
with timc, thcrc is a strong indication that the model
doc!; nol dc~cribe thc reservoir adequately (i .e., the

~'cll prodllccd only oil I1lld di~~olvcd ga~. fkforc
shut-In, a total of 14,206 SloB oil had bccn produccd.
Analysis of these data show that afternow distorts
none of the data recorded at shut-in times of I hour
or f!1°re. B?sed ?n the slope .(~50 psi/cycle) ef the
earliest straight II~e, p~rm:ablhty, k, a~pe~rs to be
30 md. Depth of Investigation at a shut-In tIme of I
h~ur is 1.44ft: lending confidenc~ to ~hechoic~ of the
middle-time line. Pseudoproduclng time, tp' IS279.2
hours.
From these data, determine whether the buildup
testdata.indicate that the w~ll is beh,avingas if it were
near a single fault ~nd estImate distance to an apparent fault from buildup data at several times in the
LTR.~-1\'c~I..i~

not .~having a~ if it ~ere in a rcscrvoirof
unllmm thlckncss and porosity, and much nearer
oll,c.houndary t-,lan any others):
,
IIIC fulluwlng cxamplc Illustratcs this compu tatlona
.
I tecI1l1lque.
.M~

S()luli()n.-Our attack is to plot/?~ VSi(/p+At)/~t;
extrapolate the middle-time line on into the L TR;
rcad pre~sures, PMT' from this extrapolated line;
sublraclthosc Prcssurcs from obscrved values of /J
in the LTR (Ap~ =Pws -PMT); estimate values of

kh

1

E.\'Ol11ple 2.8 -Estinloting

Distance

toaNo-Flo},'BoUlldary
(Jr()I)frm. Geologists suspect a fault near a newly

L from Eq. 2.28; and assume that, if calculated
va~uesof L ~re fairly constant, the well is indeed near
a single sealing fault.
From Fig. 2.21, we obtain the data in Table 2.7.
We now estimate L from Eq. 2,28. Note that

PRESSURE

BUILDUP

TESTS

43

TABLE
3,792

4>,i.tC,

=

(3,792)(0.15)(0.6)(

1.7

k

x

10 -S)

30

~t

=1.934xI0-4.

We

first

that

estimate

the

L

At

=

approximation

10

I

hours,

=1

+

.P
thts

which
Al

is

assumes

adequate

in

p.

Ap;"s

= 52

=

-70.6~

(

Ei

'

792

kh

<f>ilC L 2

'

)

kAI

-(70.6)(1,221)(1.310)(0.6)
=
(30)(8)
2

(

-3,792

(P..~ -PMT) = ~p:",

(t"t~tl/~t

(psi)

(psi)

6

475

3.996

3.980

(PSI)
16

8

359

4.085

4.051

34

10

289

4.172

4,120

52

243

4240

4,170

70

14

209

4:298

4,210

88

16

18.5

4.353

4.250

103

20

15.0

4,435

4,300

135

24

126

4.520

4.355

165

30

10.3

204

4,614

4,410

36

8 76

4,700

4,455

245

42
48

7.65
682

4,770
4,827

4,495
4,525

275
302
330

54

6.17

4,882

4,552

60

565

4,931

4.578

353

66

523

4,975

4,600

375

).

q"i.tC(L

'Ei

PMT'

FROM

12

case.
B-3

VSIS OF DATA
BOUNDARY

P..s

(hours)

at

2.7 -ANAL
WELL
NEAR

kAt

.

,
I

-. (

-1.934x

EI

10-4

) -0.184.
-

L2

10

"

Sl~
2
L

(1.107)(10)

.1300 poi"C~"'\v/,,1

4

=

4
1.934x

.

"
""
/

=5.72xIO,

.<1;

10-

C-

or

I.n

~
L=239

ft.

-.11

For
larger
values
of
I p = I p + At
becomes
terms

in

Eq.

satisfies

the

2.25

can

equation

boundary

be

for

for
the
case
In
has time
to double,
to

shut-in
time,
decreasingly
neglected,

all

values

which
the
estimation

slope

is easier.
time,
intersect

Al x'
(Fig.

distance

L

the

but
of

the

no

L =240

shut-in

of the
distance

of

From

find
the
sections

from

the approximation
accurate,
and

ft

time.

KX)

buildup
from

buildup

tests

test
well

plot,

to

the

fault

can

the

I
tpt

we

at which
the
two
straight~line
2.22).
Gray
14 suggests
that
well

cycle

~

t

-"KtFig.

2.21

-Estimating

distance

to a no-flow

boundary.

be calculated

from

L=J~~~~~~~~!.!.

(2.29)

ct>IJ.C,
In

Fig.

2.21,

shows

in

slope

did

double,

and

(/p+Alx)/A/_,,=17,

AI x = 17.45
ft,

the

that
hours.

Eq.

reasonable

the

figure

from

2.29

then

shows

agreement

which

that

with

L = 225

our

previous

P ws

calculation.
The

results

be used

to

compare
after

of

average

of

(barrels),

AN

produced
the

p

If
is

between

average

then

the

sometimes

The

basic

pressure
quantity

VR

Times

reservoir

production,

reservoir

reservoir,

c,.

tests

size.

a known

volumetric

pressibility,

buildup

reservoir
static

production

closed,

pressure

estimate

the

constant

pressures

a material

and

PI

before
balance

u

t

X

a

com-

and

and
on

I

A

volume

barrels
2,

log

and
from

reservoir

stock-tank
I and

fluid

lp+~lx

is to

before

of

with
is

can

idea

the

of

oil

P2

are

after

t
log

oil

+

i1t

p
i1t

,

reservoir

shows that
P2

-p

I

(ANp)
(Bo)
V Rc «f>

'

Fig.

2.22 -Dislance

to boundary

f,om

slope

doubling.

...

or

pressed in thousands of standard cubic feet per day
(~N ) (8 )
J-'R= '~'-J1'-'~o'.
(p, ~P2)Ct(/)

,..,..,.,.(2.30)

(Mscf/D), and gas fC?rmation volume factor, Bg, is
then expressed in reservoir barrels per thousand
standard cubic feet (RD/M~cf), so that the product

Exailiple 2.9- Estimating
Reservoir Size
I'rcthltm. Two pressure buildup tests are run on the
only well in a closed reservoir. The first test indicates
. the Second I'nd,' cates
an average pressure 0,f 3 000 pSI,
2,1()() psi. The well produced an average or 150
STB/lJ of oil in the year between tests. Average oil
furlllationvolumcfaclor,Bo,isl,3RD/STD;total
comprcssibility, ct' is 10 x 10 -6 psi -I; porosity, (/),
i~ 22%; and avcragc ~and thicknc~s, h, i~ 10 fl.

q~ B~ is in reservoir barrels per day (RB/D) as in the
analogous equation for slightly compressible liquids.
2, ,A}I gas prope!ties (Bg, JJ.,and ~g) are evaluated
at original re~ervolr pressure, Pi. (More gener.ally,
these pro:pertles shoul~ be eval~~t~d .at the uniform
pressure
E 2 31 In the reservoir before Initiation of flow.) In
q,.,
8 -178.1 Zj Tpsc(RU/M f)
~jpoT
sc,
I SC
C.,j=C~i'\'R+CII.SII,+cf~C~j'\"~.

Eslilllalcarca,AR,ofthcrcscrvoirinacrcs.
Solution. From Eq. 2.30,

3 Th f
D .
.e
actor
IS a measure 0 f non- Oarcyor
turbulent pressure loss (i.e., a pressure drop in ad.

AN B
--.p-~o
(p I -P2)C, (/)

dilion to that predicted by Darcy's law). It cannot be
calculated separately from the skin factor from a
single buildup or drawdown test; thus, the concept of

(q/)B 0
(p I -P2 Xc t (/)

apparent skin factor, S'=s+Dq",
is sometimes
convenient since it can be determined from a single
test.
For many cases at pressures below 2,CXX>
psi, flow

VR =

=

in an infinite acting reservoir can be modeled by
=

(ISOSTB/O)(365daYS)(I.3RB/STB).
(3,<XXJ-2,IOO)psi(IOx 10-6 psi-J)(0.22)

2 = .2+ 1,637q"Jl.iZ;!
Pwf P,
kh

Thus,
J'H = 3S.9x 106 bbl

( 1,688 (/)JlCti) -<::!+Dqg

./

llog

kIp

1.151

)

].

= 43,560ARh ,

(2.32)

5.615
anI.!

6

superposition
Using these basic
to develop
drawdown
equations
equations,
describing
we can use
a

,.1 = (35.9 x 10 bbl)(5.615 cu ft/bbl)
H
(10 ft)(43.56 x 10.1sq ft/acre)

buildup test for gas wells.
Forp> 3,000 psi,
q 8

=463acrcs.
2.11 Modificariol1s

P,'s=p;-162.6'fg-glr-1
I
kh
anI.!

for Gases

l11is scclioll prcscllts modifications of the basic
ura\\uo\vn anI.! buildup equations so tha~ they can be
applicl.! to analysis of gas reservoirs. These
Illuuificalions are based on results obtained wilh the
ga~ pscudopressure.IS although a more complete
uiscll~sionoflllatsubjcctislerltoChap.S.
Wattcnbarger and Ramey 16 have sho\vn that for
some gascs at pressures above 3,000 psi, flow in an
in finitc-act illg reservoir can be modeled accurately by
tile cqualion

[ 1.og

( 1,~88q,Jl.;~!;)
kIp

.k

.

~)+3.231.
(/)Jl.;Ct;II'
For P < 2,000 psi,
q Jl.-z. T
-1,637

'fgr"'1

'log

(2.34)

( 'P
I +, -.,
~/ )

...(2.35)

~I

and

---

2
2
) = 1.151 (PI hr -Pwf)
g

1.151

..(2.33)

k

s' =s+D(q
__(S+DQg~',

~I

-Iog(

=pf

.kh

,log( I'/1'+ ~I
-., )1

(p
-P
)
= 1,ISll \PI hr -Pwfl
111

s' =s+D(qg)

-P~I'S

-162.6QgBg;Jl.;
Pll'r = P,+
.kh

-JJ..

l

mN

(2.31)

This equation has the same form as the equation for
a slightly compressible liquid, but there are some
important diffcrcnccs:
I. Ga~ production rate, q~, is conveniently ex-

-log(

.£... -_2) + 3.23],
(2.36)
q,p.iCt;' w
where /11N is the slope of the plot p~
Ys.
log(tp+~t)/~/J,whichisl,637qgJl.;z;T/kh.

:
~

An obvious question is, what technique shollld be
used to analyze gas reservoirs with pressures in the
range 2,(XX)<p<3,(XX) psi? One approach is to use
equalions writlen in terms of the ga~ pseudopre~~ure
inslead of either pressure or pressure squared. T!li~ is

c=('
II

S =(3.44x
g/ g

10-4)(0.7)

=2.41 x 10-4 psi-I,
alld

at least somewhat inconvenient, so an alternalive,
approach is to use eqllalions wrillen in terms or

I ~PI hrIII-PHi)

s =s+D(qg)=1.151

either Pws or P~s and accepl the resultant inaccuracies, which, in real, heterogeneous reservoirs,
may be far from the most significant oversimplification on which the te~t analy~is procedure is
based. The smaller the pressure drawdown during Ihe

( ~;7?k

-log

) + 3.23 I

/ II H'

le~l, Ihe less the inaccuracy in this approach.
~(2,525 -1,801)

Example 2. 10Test Analysis

= 1.151t

Gas Well Buildup

l

?96.x 10 -4)(0.3)2
-lo' g (0. .18)(0.028)(2.41

lest.
Test data include the following.
l)rctltlem.Agaswclliss!luliI1roraprcssurchuildup
qg
Mscf/D,
T =
= 5,256
181°F=64I°R,
h
Ili
Sw
~

=
=
=
=

81

~

I

+ 3...23J = 4 84

28 ft,
0.028 cp,
0.3,
0.18,

From results of the P~vsplot,
T
k= I 637 q_~Jl.iZj.
,
mwh

Zj = 0.85,
r w = 0.3 ft,
Cgj = 0.344 x ~O-J psi-I,

(1,637)(5,256)(0.028)(0.85)(641)
(4.8x 105)(28)

=

I

Pi = 2,906 psla, and
Pwf = 1,801 psia: -=9.77md,
Most of the test data fall in the intermediate pressure
range, 2,(XX)<p<3,<XX> psia. On a plot of Pws vs.log
(tp+At)/~t,
t~e MTR had a slope, m, of 81
psI/cycle; on this plot, PI hr was found to be 2,525
2
I
.. Alternatively, on a plot 0f Pws
psla.
vs. °g
(ti +~/)/~/,
the MTR had a sloRe of 0.48x 10
psl1/cycleandprhr of.7.29 x 106 psi2.
From these data, estimate apparent values of k and
s' (I) based on characteristics of the P ws plot and (2)
based on characteristics of the P~s plot.

d
an

2

'=1151
s.

2

[ (Plhr-Pwf)

-IOg-~+3.23
~JI.-c
,I.r2.H
/

mw

1

6"
= I 151f E.29x 10 -(1_,801)-]
.t
4.8 x 105
-10

.g
Solullon. From results of the Pws plot, for standard
conditionsofI4.7psiaand60°F,

I

I 3..23!

9.77
--+
(0.18)(0.028)(2.41 x 10-4)(0.3)2

..

=4.27.

8 .=178.1~~
gl
Pi Tsc

Neilll\.'r ~ct of r\.'slllts (k and ~") is ncc\.'~~arilymm\.'
accurate than the other in the general case; as in t!lis
particular case: use of an analysis procedure based on
(178.1)(0.85)(64I)(J4.7) .~as.pseudopress~lre
can be used 10 improv~ accura~y
=
If dl~agreement In results from P..'s and PHosplots IS
(2,906)(520)
=

0.944

unacceptably

large.

RB/Mscf,

Iii

...'
~.~~-~-~

B ..Uasi..:
k= 162.6 qg-gIJl./
mh

2.12
i

Modifications

for

Multlphase

!
-1

Flow

buildup and dra,,:down cqualions can be
modified to model multlphase flow.17.18 For an
infinite-acting reservoir, the drawdown equationI
becomes

= ~162.6)(5,256)(0.944)(0.028)
(8 J)(28)
,.

=9.96 md,

p

( ! ,68~q,c,r~I:

=p. + 162.6 ~ [IOg
wf,
)., h
--,

s

.,

1

1.151

I

}., I

)

.

(2.37)

1

!

alld I Ilc buildup cljual iOll bccomc~
q Rt
I +, -.
AI
p",~ =Pi -;- 162.6 -10g(.p
Ath

AI

) .,.',.,

(2,38)

In the~e equations, tlte total nO\\' rale qRt is in
rc~cr\'oir barrels per day (neglecting solulion gas
liberated from produced water)
R '
l/Rt =l/II/l'1 +(l/.1' -.~:~-

)8.1' +(111,1111"

E.\"a/l1p/e 2. II -;\-Ili/tip/lase
Test Ana/ y sls

Blii/dlfp

P,roblem. A buildup t,estis run in a well that prod\ll:t'\
Oll~water, an~ gas simultaneously, Well, rock, an,d
nuld propertIes evaluated at average reservoIr
pressure during the test include the following.
SO = 0.58,
5,f = 0,08,

,..,."
I

.1

allutola

,. mo

-A:!I

b . I '~.
I Ily,

(2.39)

"t,IS

~-".

A:8

A, -+

+

,.

., ...,

..,

(2.40)

IL"
ILII' It.1'
.T(ltalcomprcssibility,ct,wasdcfincdinEq.I.4.
"llc~c C4lmtion~ imply that it i~ po~~iolc to
uctcrmillc A, from the ~Iopc III of a buildup tc~t run
011 a \\'cll that prodllcc~ two or tllrcc pha~c~
~imultallcoll~ly:
q H,
A,=162.6-. Illir
,...,...(2.41)
Perrlne
.17
1las s110wn t I lat It IS aIso pOSSI
.bl e to
c~(imate thc permeability to eoch pl,ose nowing from

..

the samc slope, ",:
q B II.
kn=162.6
(} () (I,
",1,

(q~-

q(}

-~

k.e = 162.6

(2.42)

R

s

) B~II.~

,f

It,} =
11."' =
It" =
B() =
1111'=
B.1' =
R~ =
~'#' -.,
'" ' =
h =

(2.43)

1,5cp,
0.7cp,
0.03 cp,
1.3 R8/STB,
1.02 RB/STB,
1.4ROR8/M~cf,
685 scf/STB,
017

0.3ft,and
38 ft.

From plots of Bo vs. p and Rs vs. p at avcra~t'
pressure in the buildup test,
dR
-:! =0.0776 scf/STB/psi,
d
an

,

",1,

5",=0.34,
36 10 -6
'-I
c",=.x
pSI,
35
10
-6
'-I
cf = ,x
pSI,
C = 0.39 x 10 -3 psi -1 ,

lip

dBo
-=

I 6 R
2.48 x 0 -B/STO/psl.

.

dp
alld
B
"q",
A:1I,-162.6.

1I,/tll'
...,..,
(2,44)
",1,
.nlc term «(I -(loR~/ I,OOO)B~, wltich appcar~ in
1:4~, 2..19 anJ 2.4.1, i~ (lte jrc'£' ga~ now ratc in tlte
rt"'cr\'oir. It i~ f()llnd oy suotract ill!! thc di."s()I\.cd!!a~
ratc «(I"/?\/I,()()(») fronl tltc total ~urfacc ga~ ratc
«(IE,)and con\'crting t? a reservoir-condition ba~i~.
Slmullancou~ ~olutlon of Eq~, 2.37 and 2.38 re"..Its in the follo\\lng expre~sioll for tlte skin factor s.
...=1.151 /" Ihr

- ,~-log(-~)+3.23,
,
A

III

I

The production rates prior to the buildup tc~t \Vcrc
qo = 245 STB/D, q", = 38 STO/D, and q" = 4R9
Mscf/D.
A plot of P"'S vs. log (I l' + At) / ~t shows tltat tht'
slope of tlte ~TR, "'~ IS 78 psI/cycle and tl~:lt
1'1 hr = 2,466,psla. FlowIng .pressure,P'vf' at thc In.
sta~lt of shut-In wa~2,~28 psla.
l'romtltc~cdata,cstlmatcA"ko.k""k-i,and\",
Sfllution. Permeabilities to each phase can hl'
determined from tlte slope ", of the MTR:

k(} =162.6

q0 8 011.0
",II

1/>£." II'

(2.45)

(162.6)(245)(1.5)( 1.3)
==26.2md,
(78)(38)

Stat ic drainage-area pressure, p, is calculated just
a~ for a single-pl\a~e reservoir. In use of the MBH
charts to determine p (and in the Horner plot itsel0,

kw=162.6

thc effective production time tp is best e~timated by
dividing cumulative oil prOduction by the oil
prodllction rate just before shut-in.
An important. assumption required for accurate

=

use of these equations for multiphase now analysis is
that saturations of each phase remain essentially
uniform Ihroughout the drainage area of the tested
\vcll.

q",8".11.'"
h
",

(162.6)(38)(0.7)(1.02)
78 38
= J.49 md,
( )( )

q R
(q" --fiiii
k = 162.6
'
"
",h

)BgII.g

PRESSURE

BUILDUP

TESTS

I489

-47

I

(245)(6X5) (1.480)(0.03)

= 162.6

(1,000)
(78)(38)

= 0 782 md
..made
To calculate total mobilit
A we first need lotal
now rale.
y, I'
, q RI'
=
B +
-(} q RS )B +
qRr qo 0
qg
I (XX)
g q..,BI"
,

(

= (245)( 1.3) + [489 -(245)(685)
1,<XX>

2. I. In Examplc 2. I, ~'hat error arises h\.'I:"II~C:\\'\.'
II~cd Eq, 2.4 to call:ulate skin'fador
il1stc"d or thc
more e,\"ct Eq. 2.3? What dirfercnce would it Ilavc:
in the value of s had we used a shut-in timc: or
10 hours in Eq. 2.3. and lIte corre~ponJing v~lue or
P\l'~? What assllmptlon have we made about dlslanl:c
rrom lested well 10 reservoir boundaric:s in Exampll.'
_..
') I ?
2.2. I)rove thallhe slope of a plol of shut-in HI IP
vs. log (/,,+.1/)/011
is, as asserted inlhe
lext, Illc
difference in pressure al two points one cycle apart,
Also prove that, for .11 cC1P' w~ obtain Ihe same slope
on a plot ofPM'S vs. log dr. Finally, prove that on a
plot of Ph'S vs. log ~/, we obtain the same slope
regardless. of Ihe units used for shut-in t!me, .~, on
the plot (I.e., that ~I can be expresscd In mInutes,
hours, or days \vithout affecting the slope of Ihl.'

]

.(1.480)+(38)(1.02)
= 833 RB/D.

plOI),
2.3. A well produl:ing only oil and dissolved gas
has produced 12,173 STD. The well has not been

Then,
A = 162 6 ~
I
'mh

= ~~~~
(78)(38)

stimul"tcd,
nor is there any reason to
thl'rc: is a signifil:ant
amount of formation

=45.7 md/cp.
To calculate skin factor, S, we first need Co and C,:
-~

c-0

1;~"
.,'~ r " .I""":
, ,

dRs
dp

Bo

.I

dBo
dp

B
0

-CI

= (1.480 RB/Mscf)(0.0776
scf)
(1.3 RB/STB)(STB-psi)

.1

Mscf

-(2.48

1,<XX>scf

x 10-6)
1.3 (psi)

believe that
damagc.
A

pressure buildup test is run with lIte primary objeclive of estimating static drainage-area pressure.
During buildup, there is a rising liquid level in the
wellbore. Well and reservoir data are:

I

cf>= 0.14,
Jl. = 0.55 CPt
'-1
= 16 X 10 -6 pSI,
r w = 0.5 ft,
A wb = 0.0218 sq ft,
r e = 1.320 ft (well centered in cylindrical
drainage area),
p = 54.8 Ibm/cu ft,
q = 988 STB/D.

i
I
!

'
1

B=I.126RB/STB,and
h = 7 ft.

I
'

= 86.4 x 10 -6 psi -I.
Data recorded during the buildup test are given in
Table 2.8. Plot PI"S vs. (lp+.1/)/~1
on semilog

Then.
c =5
+5
+5
+
r
oCo
gCg
I"CW Cf

10-6)+3.5x

x 10 -3)

10-6

the buildup t~st data..
2.4. Consider
the
Pro?lem
mation

= 860 x 10 -6
.pSI,

.-t

(

og ~.

)

AI
323 1
+ .single
I III

-log

78

[ (0.17)(86.0

~~I:ate

buildup
the

test
and

estimate

MTR

In
for-

permeability.

-

original reservoir pressureestimatesIn thesecases:
45.7
x 10 -6)(0.3)2

] + 3.23 J

(I) some LTR data were obtained, but final straight
line was not established; and (2) no L IR data were
0 b2.7.
talne.
. dConsider the buildup lest described In
Problems 2.3 and 2.4. Estimate static drainage-area

= 1.50.

I

.2.6. Provethatinab.uilduptestfora.wellneara
fault, the technique suggested In the text
(extrapolating
(he rate-time line to infinite shut-in
time) is the proper method for estimating original
re~e~voir pressu~e. Comment o~ the p~ssible errors in

= 1.151\ 2,466 -2,028

l

2.3.

..
described

2.5. Consider the buildup
test described
in
Problems 2.3 and 2.4. Calculate skin factor, s;
pressure drop across the altered zone. (i1p)s; now
efficiency. E; and effective wellbore radius. r wu.

and
m
S=I.151 [ Pthr-Pwf_1

I

paper and (PI"S -PI'f)
vs. .1/t' on log-log paper-yand
estima!e the time at \vhich aflerno\v ceased distorting

= (0.58)(86.4 x 10 -6) + (0.08)(0.39
+(0.34)(3.6x

!

pressure for this well (I) Iising thep.

.;

method. and (2)

..

.TABLE 2.8 -PRESSURE BUILDUP TEST DATA

~t

P.,

~t

Pw.

(hours)

(pSla)

(hours)

(pSia)

709

19.7

4,198

0

197
2.95
394
492
591
788
9.86
14.8

3,169
3,508
3,672
3,772
3.813
3,963
4,026
4,133

246
296
345
39.4
-144
49.3
59,1

TABLE2.9-BUILDUPTESTDATA
FORWELL NEAR FAULT

4.245
4,279
4,306
4.327
-1,343
4,356
4,375

~I
(hours)
20
30
-10
50
100
200

USilig tIle mouificu Muskal melhod,
2.8, In Exalilple 2,7, explain how we could ha\'e
applied the modified
~1uskat method to estimate
SIalic drainage-area
pressure if we had not had
eslimalesofk/lt>Jl.c,
orrt"

I

2 9
E
.
"stlmate
r
t
f
ac or
rom

i

,I
'

II
we

f
t

h

e

b
pressure

.

ormation
f II
'
0 owIng
' Id

UI

d

b 'l '
d
k .2,11,
I Ity
an
s In.
' I bl
f
aval
a
e
rom
a gas,

permea
ala

I
up

les

R~ =
It> =
r M' =
II =

~t
(hours)
500
ROO
1,000
1,500
2,000

"
well

pressure

believed
"

to

.

I

f

h

buildup

be

near

"

reservoIr,

test
was
",
a sealing
fault
EstImate

k

II

all I, given t e we ,roc,
an
anc.JthebuildupdatainTable2,9.

T = 199°F=659°R,

Pw~
(pSI)
2,225
2,360
2,434
2,545
2,616

748 scf/STB,
0,18,
0,3 ft, and
33 fl,
A

Infinite-acting
,

Pw~
(pSI)
1.373
1,467
1,533
1,585
1,752
1,940

d

n UI'd

the

run
In

on

an

,

an

011

otherwIse

dIstance

to

'
propertIes

the

below

II = 34 ft,
Jl.i =
Sit' =
('!!; =
It> =
z; =
rlt, =
i

0.023 cp,
0.33 (water is immobile),
0.000315 psi -t ,
0.22,
0.87, and
0.3 ft.

q =
Jl. =
It> =
CI =
h =
P; =

Np = 84,500 51'0,

A

..

1hc wcll produccu 6,068 Mcl/O bcfore the te~l.
plot ()f IIIIIJ, PIt~' v~, log (I'I+~/)/~I
gavc a
miuulc-time
line with a slope of 66 psi/cycle.
An~lysis of the buildup. curve showed. that static
uralnage-~rea p~essur,e, p, was 3,171 psla. Pressure
ou thc mluulc-tlme
line at ~I = I hour, PI hr' wa~
2,745 psia; nowing pressure al shut-in, Plti'
wa~

940 STB/D,
50 cp,
0,2,
78xl0-6
psi-I,
195 ft,
2,945 p~i,

I

/1

= I,ll RB/STII,
(I < 20 hours
,,'bs
.

Bo =
P; =
Jl. =
k =
S =
II =
c, =
It> =

~9 psi/cycle,

r It' = 0.333 ft.

Flowing

prcssllre at shill-in,

Plti'

wa~

slrniglll

lillcal~/=lhour,Ptllr.'\a~I,744psia,PlotsofB(}

anu

2.12. ' A well nowed for 10 days at 350 STB/D; it
was then shut in for a pressure buildup test. Rock,
" nuid and well properties include the following
'
,

2,4R6p~ia.
2,10, Estimate total mobility,
X" oil, water anu
ga~ pcrmeabilities,
alld skin factor for a well thaI
proullceu oil, waler, alld ga.~simultaneously before a
press\lrc buildup lest, Production
rates before the
test wcre qo=276
STB/D, qlt,=68
STB/O,
and
({,f =689 Mcf/O.
A plot ofplt'.~ VS, log (I" +~/)/~I
sho,,'cu tllal the slope m of tile middle-time line was
1,5RI p~ia; Illc prC~Sllrc 011 thc middlc-liltlc

.

)
(a

1.13 RB/STB,
3,000 psi,
0,5 cp,
25 md,
0,
50ft,
-6
'-I
20x 10
pSI,
0,16, and

De

'
te~mlnean

d

I
P,otl,

..

h

d
epressure

'
ISri

t

' b

'

,

utlonln

V~. P and R~ VS, P ~ho"cu thaI dR~/dp=0.263
~cf/S1'B/psi
' and
that
dB(}/dp=O,248
x 10-"

Ihe reservOIr for ~h~I-!n tln~es of 0,0:1,
uaY~,(~~Sllmeanlllflnll.eactlll,grese~VOI!,)

RO/51'0/psi,
Rock,
cludc the following,

(b) Calculate the radius ?f Investigation al ~,I, I,
and 1.0days, Compare ri with the depth to which the
tranSient appears to have moved on the plots
prepared in Part a,
2,13, In Example 2,6, jJ was determined
to be
4,411 psi, Bolh the Horner plot and the abscissa of
tlJe MOH chart used tp = 13,630 hours. It can be
shown that for a well centered in a square drainage

'~f)
'~.l'
")'It'
('It.
,. c{

r

=
=
=
=
=

nuid, anu well properties

0.56,
0.09,
0,35,
3.5 x 10 -6 p~i -I,
3.5 x 10 -6 psi-I,

c'!! = 0.48 x 10 -3 psi -I ,
1«(1 =
It", =
It.~ =
Bf) =
Bit, =
IJ" =

1.lcp,
0.6 cp,
0.026 cp,
1.28 R8/STB,
1.022 RB/STB,
1,122 R B/M~cf,

-~

in-

I, and 10

area, the time required to reach semisleady state is
tp.t~=(It>p.cIA/O.000264
k)(IDA)p.u
and
that
(' /)/1) s = 0.1. Show that if tp,t\' is used instead of t p
in boththe
Horner plot and in the abscissa of tne
MBH chart, the resulting estimate of fi is essentially
unchanged, Buildup data (from the MTR only) are
given in Table 2.10. Other data include:


Related documents


PDF Document parametri al study on retrograde gas reservoir behavior
PDF Document soil mechanics
PDF Document welltestgasoilh2o
PDF Document 61 me porosity permeability skin factor
PDF Document 28i14 ijaet0514243 v6 iss2 804to811
PDF Document civil sem6 ah ii assignments


Related keywords